NumberDB$^\beta$ Advanced search Tags Tables Help Login
Values of the Clausen functions $\mathrm{Cl}_s$ at rational multiples of $\pi$
edit · history · discussion · files · long url · special functions period special values
Numbers
$s$
$t$ 
2
1/2:
0.9159655941772190150546035149323841107741493742816721342664981196217630197762547694793565129261151062
comment: Catalan's constant $G$.
2
1/3:
1.014941606409653625021202554274520285941689307530299792017489106776597476258244022136470354228256695
comment: Gieseking's constant, the volume of the regular ideal tetrahedron.
2
2/3:
0.6766277376064357500141350361830135239611262050201998613449927378510649841721626814243135694855044633
comment: $2/3$ of Gieseking's constant.
2
1/4:
0.9818721510502033567179245430601956671307909716607304615766131346531556650497696362249028028843877241
2
3/4:
0.5238893539615938491906227855940036117437162845198943944433640748422741551616422514852245464213301710
2
1/5:
0.9237551681005353087119860297930243539662790064125172517077128483251509833253097572872832218011226000
2
2/5:
0.9973546913984147786672835753214826444165571661894434204718089638932886257789849954087218674412963266
2
3/5:
0.7848157801977508189781114542553411285375569545305456497359047807040352905610763656172607234010591083
2
4/5:
0.4250778224013279193783442421322830317580004233177955414718083663785066704358172595829222880804744367
2
1/6:
0.8643791310538927496250363151902194786681885764036897041820376897753247155820641851870219305078075779
2
5/6:
0.3569083278490659371144350380529593356973439226385398081732931363870259774529421741187867533936792304
2
1/7:
0.8096245973109173006176003522449767394530557374767106912429827031283773924840929447873546587312698297
2
2/7:
1.004653150539948718044867981322300931670272015682207148994394146989756364398115138269414580418607208
2
3/7:
0.9801480444923570033745743968309527896699698008012832583489538993551915872669301073883063101954404896
2
4/7:
0.8264990334718855325769912160390396528610099359834796999760610845384419821244124195619826259344573768
2
5/7:
0.5914036338040059517563723733027811052397670476904672990063636047205353733359089284884232674513785197
2
6/7:
0.3072980220409429415951663615838262736179197296356071167457856296334992102850353756526473685219662257
2
1/8:
0.7606012393584689884286131625720173426893798177524510966638111574524861535634952998185973392247719279
2
3/8:
1.007876162955838197171789451779492400702061871051578680834542639193826186728000205119670542620866849
2
5/8:
0.7459314859750412725764780589824905948302037287916314836128606017726891091471790793770582694102017637
2
7/8:
0.2696651638333673100696508910419195091239843319220858658755045901259083210386104817061459377825780658
2
1/9:
0.7170468960853112497480248794102429340718650985729127564332731356861206826555997343951979999487718529
2
2/9:
0.9537410166388513170665182230590592662495397623871479685828659079138492976685161948447740386685578460
2
4/9:
0.9683748252025916582765723122121013925429262447597534536097085103593570034324702713428138294545492882
2
5/9:
0.8482866313196488626691894282717447420693786360700840675387884194947589865217994528564083391473028232
2
7/9:
0.4695536040375554879282320669530085699780766400072712417780116527341707959522810591733671239412832019
2
8/9:
0.2401763877658855912147657678807133009470952173793387721418401817291960338213416369728109806144929300
2
1/10:
0.6783410621110970244312541924128488339010749476006567531434229185026073103045639700157184791578355964
2
3/10:
1.009992452806288508019626954501381671853554184342832706357070149446033595024007529889955459728265766
2
7/10:
0.6175845627074130985305712273737111075847757070775598814891177590940159497434693470813250980277362118
2
9/10:
0.2164634780608293700752611775163366569179354443943981272895664943400318186419090913720768682572742964
2
1/11:
0.6438262100584401303611066755032692806849826541852182493029198451516672312870493018370816241228390865
2
2/11:
0.8936729400529264257948818880761383784689145397699422348783702380776022377103006441793796650986685421
2
3/11:
0.9979869190156514728495121697525529889267240328099220865667216290084843588303294481111219000928283157
2
4/11:
1.011155392069913374817081038174013429591555517612206406508563310207536082501757582150160383748366557
2
5/11:
0.9601064004695693005243917514869897838767140027818757612203523581676978975122232587126658023565734889
2
6/11:
0.8616115304535808417925588857543897381514513106317521952884849951112648412962737688389699065698210812
2
7/11:
0.7275648151754828488404646747363343696660563288651834120473237444811101134106613003043419103444076692
2
8/11:
0.5671811537888610519532327268753581198509983774940459889224791314528519381821925636916369468079177751
2
9/11:
0.3880952440179697383863413689891316636731367809638390316240885829738341964594218531042994732244852637
2
10/11:
0.1969897400319769174636657314652000914505253843002471318637347261128661124318989797473917915735048155
2
1/12:
0.6129061441556218400784452246529219807884900720534323696821369745760760257499689445851513522342185044
2
5/12:
0.9879589450250142055869773482533400380039479940056865143532765727274779479548257377214683410575923825
2
7/12:
0.8095047811004812370297598292268603701552760326864166102666300045339649592283546506620749643607527673
2
11/12:
0.1807165786286754652659270670578122414543957838515875175911181296884136679589368519916403869803147154
2
1/13:
0.5850686644998759857239673261618879099884422267310167827987225758709458035730936683246910681651391004
2
2/13:
0.8363018901407091390335172092988849126210807680736689463299680570217405691976474297285403356342745562
2
3/13:
0.9634490042365788097095445487210371305323509506465836982641060277798660389589068017176916939161052470
2
4/13:
1.012041528824899919719085359268818874135684889106836442936997815390116697065722652692281143008415227
2
5/13:
1.004354940555469707242244344445741160084136478903139935735074380991618083563769099391974565579681992
2
6/13:
0.9540405707291433927115690028838330217007846749460725620371970074829411167073373509333508789202610132
2
7/13:
0.8705817110143826846079646421276102948618337535989814072203277338029033572202023742031404287462601020
2
8/13:
0.7611405811194661505653643208061808502431571723163662271123206189724203432611500362664664383516946219
2
9/13:
0.6314712382651668444364031988657284490141063029486533293808375059039065254351476345590479238325679160
2
10/13:
0.4864287188720071133537600472791206196819586131735474172455075240383954806052381262510162544559747404
2
11/13:
0.3302811257282591791739745296644754755532383235202507248614691493266822206647861033823997641300669427
2
12/13:
0.1669177194295214162072087215124454536779018426941823096337385473600755189742699534604209003480018223
2
1/14:
0.5598827488059319285384753267295734638156126320920034103175380982285853767858199403072102245408267732
2
3/14:
0.9438402705237005250602372298995212236027526004130002546374328979586084080793959879113155004039775294
2
5/14:
1.012586479331821024517099208167654229699509999716283175447277541637317908661132428778462615817732782
2
9/14:
0.7168846624298180486389130215162636770796264758710495259440957392770502219931779645342509820920435220
2
11/14:
0.4537662482775220233729500314840448287677677000123586254629559482810126144459309342171623453062572846
2
13/14:
0.1550704501504732782296751506070850940890847633536480646960467466643966805437734679135328951751918583
2
1/15:
0.5369881772587339233550764490227148564197348805720671676114828833969956774219957623821725507210625426
2
2/15:
0.7843979708739228119442910843813080206306242909208330313722318928993336802440388471011788131418924150
2
4/15:
0.9942581070437515677438007492001828088981345721837246157037114824054620119576144097714461526513381370
2
7/15:
0.9494188253866653971250062548224605939686503597313060919827874064537418940701826109636520508169929104
2
8/15:
0.8770242294757791384335408014064301709164389921754807660201039379800774754201944415478604787419347428
2
11/15:
0.5557459923058619985270303484969677234399150760959842326936595134154232742475171889975159132803707656
2
13/15:
0.2872689173520470280723907097812166161815570048289707235203761516966026742652316422154557368162233465
2
14/15:
0.1447891918217725173829309068320608461044227351116506519253669369473288372999763388315831441501163351
2
1/16:
0.5160840336035987923756557638301671865322103636638497387463441869313783502978776779526237947166393876
2
3/16:
0.9036442262105757247181130684153867899388848678964306941980737325603433987292600906258614181582993442
2
5/16:
1.013038613252429268653365870923830862568032226667502896591138156775293650950415377349544742012200363
2
7/16:
0.9737124334142727301663115361275108488193349212975618111338680999762123638180358044483896400720823471
2
9/16:
0.8388798514975890751314860906065510942573427553365188781961158049132582032987305635953166711807933142
2
11/16:
0.6400728702649086323651268414325855651529303622716871547847078558889490963768258376610156073070994808
2
13/16:
0.3997061447326566261322183425256405895878539323706413537808024129634303053652599880660261468478659196
2
15/16:
0.1357834139243642981613491825441585151875204547876241904144386082051352735161300280433251251042534236
2
1/17:
0.4969182499647387012468047615104531464275982014690089211504736311707321260633060064316639234308700472
2
2/17:
0.7381762731461587189603194016228755384201659216280240926758678426383605274279364271697824218873839712
2
3/17:
0.8837928154874576957505902368611542201106462244997280008506237964124117414396936513177195104070966179
2
4/17:
0.9682011899811139806421298670016354977801199943087480958641623055704185671385142146650029608805220922
2
5/17:
1.008039699084622507442670518779611757125762632977531672619250400673769258854867816227181460940495936
2
6/17:
1.013336475068157575448283915497327641765522838550029313628888293024370708781491913722908033810146574
2
7/17:
0.9909122599535947610265074239994640807917941706420755724814447676632457634001155264283027405870880194
2
8/17:
0.9457871717455097740094134804573524059743809251678403948152062545021696113811868857294034746375730062
2
9/17:
0.8818721150496801031260909501078447173656233048403419574089363995763936802065179893060171183939839754
2
10/17:
0.8023499709769053070132832844432188811778517680297189004633549427131325698756416792001699938360763539
2
11/17:
0.7099041182700726484802694772183264834971044640686932024351018283657692218229684254093598017989176940
2
12/17:
0.6068647135961698539360288765580023165368367489626722223875729293172029739170469766270964640224577595
2
13/17:
0.4953076041083590936374231267729592947929295317248278984565591783193337614479207718003012235617355890
2
14/17:
0.3771245779533789080264482791124903992278848052247133440361796499002263870489476944562654935020233310
2
15/17:
0.2540756781556017286392544681220577895301059244736500447437866898531512438586793198372809414471229251
2
16/17:
0.1278301133916593417666450606990153772175152406549968748125397098515518623493377928467727124871780617
2
1/18:
0.4792787600752436680630911513136517073583272470166815935167943428476170459758119977638848189842044269
2
5/18:
1.000790624508572847791861019049961937648482583541576685720476039186369350769504662018559095793271985
2
7/18:
1.002576493924805578609461384640839066071762640392288631629536102061917396266536067878582663671864076
2
11/18:
0.7677996919060278346453453511643347810827243203886530107405302756948319982903955382918991017012224748
2
13/18:
0.5766473088487484164572663049140895666137932655065346519510818294389898575086049355903549262196205738
2
17/18:
0.1207553120325880431890787116085302403223946977302252153001577750045567046480121305662858190098185004
2
1/19:
0.4629862420540806647153079248355470122481135775373326437493273691327193688651885925248066979701552666
2
2/19:
0.6971298638333846305354921615678023152369695184432785742373028439561028103373933859238147449567306530
2
3/19:
0.8455109550822491892811396633792574001822177895455231373163151873994594363582710150662946706959564513
2
4/19:
0.9388425161580646576304862995688200520089843036749548737372833540180130881298604192457555673537317166
2
5/19:
0.9919138993767518772570427066848426446614207634486313540576295537641123408492799854513773915140659940
2
6/19:
1.013597635722515485528587894116901084606159832283097199697972450349746013812340206968543575421682663
2
7/19:
1.009893307591966441293620552485223794621916351949135323841232174579492109835543760825811962243735076
2
8/19:
0.9851854373898225416721244912916485601856287715874552558026548185641012760347805284932753499597999317
2
9/19:
0.9428612208287231145680606285392556726460935851588548133270228703765364214608081719542428466522880554
2
10/19:
0.8856505657600289398442797065134327453312791760010081350116848967992024396125622221727931839063930854
2
11/19:
0.8158293687793268184137016331312451312460598348854679870689903374518080613087300727022639084672423718
2
12/19:
0.7353489993435977376261691257711606586240257642250716805653386218972365493140443236433215624633003505
2
13/19:
0.6459231360507166167155033312313207552941469501705613594153031394011277391553180451468827941900324877
2
14/19:
0.5490886164967374073349028534281262719957811754481272865517871053645111210429988743649807995608694513
2
15/19:
0.4462497974631533867944240539229957719161699178812272458359559447359624501124701549991178923738317508
2
16/19:
0.3387121372209914465168457163208068578791378734039745374673289622245864294521009115820228829851151198
2
17/19:
0.2277086057543523017202490117833922892324773666058011373686611669470962662724631763009369612798647947
2
18/19:
0.1144213101373883494475618440516458546296288183156933566306759471546679636964918995628993254917899401
2
1/20:
0.4478882448133546479237104140551928041105033719754462686183497797704848282605339054184091060252386098
2
3/20:
0.8272494162748458768659725485460374112063256117417666811818293195757181269111662880210111211242313659
2
7/20:
1.013777875538938685766520601604443543539549113144011638771823760843163596602368350369438065249775153
2
9/20:
0.9639002250255169355208588104599469093193717004502555289088664379928827271142681006052006749626457765
2
11/20:
0.8556684859951022504832282217017785808604039782530564652640831908228668177933135549191622408340086283
2
13/20:
0.7049855941852321365012349879175879897471612596052316980272648812961556217306336768287755162359070475
2
17/20:
0.3222531898717016228561590712953465752795485195703503280032942448527013293991625230760333912600984829
2
19/20:
0.1087177137578061357080833178487683871599658981751178920466383205191811731082519204105498664463208116
2
1/21:
0.4338544572063114183279061313811093231957367496196671398273083177733093270108763042069148922904968073
2
2/21:
0.6605989232622273004774123038547837551283217735664574717417699344445086296329464625689481412261477414
2
4/21:
0.9086565926165280621889943604730908596486640263159621262121107737074587175166299119025214699250498127
2
5/21:
0.9710119621397290512437833976469748327792357229562959493820272200045907741709349102667599930822771611
2
8/21:
1.005775472101498838935679173581409619429207968640109847368888954329644973164173085974306279255374736
2
10/21:
0.9404549368335091595918303167105705823129137772904093429658760418281439337601944810345079631355302462
2
11/21:
0.8886774390459102755472303269837118594971258458721901409876643665526164276629929445732875522968187779
2
13/21:
0.7553832252400116032117450539355648486178185514845642084193443547843855695187542510995532734981187171
2
16/21:
0.5007844937229744714478682392916895416227788343110912778990891990905188072908376697495060115145120380
2
17/21:
0.4057688565657786427211547736823860499340600419959072025276662965426362309345433689153683302973624447
2
19/21:
0.2062706269539632693829151236182383253039897604084764086357145475907792708746315066176874062636228351
2
20/21:
0.1035549955751977680891999794537174456315758628364384039564233505510550121944030729224408216774229366
2
1/22:
0.4207728984415861475383950162222267993650927962371537793394640285448885753513788774100201212467429776
2
3/22:
0.7926459279500647776738508625037393630812550498165049627438584049616097724703705066812961614459341780
2
5/22:
0.9595946743594207357325232894724132015639925173240917966975045731956341715155778812695716729070535672
2
7/22:
1.013942152248520305919078148929781856224570590091413065638172763744376785894306698757678493680696029
2
9/22:
0.9924571343102270567927147244837165757549714359279050478282994854145993991755580555123140039764710022
2
13/22:
0.7984095123012421875995440399891507439184030454459855320162551939276823009458471289601642673642283704
2
15/22:
0.6501597446607788814988458115616146713915424256588213596145108915038217291889760486055075385084921947
2
17/22:
0.4795414741246360854703274137289183096256355159331539160873283941117852227594662519132387717287668227
2
19/22:
0.2936524684422390412490947776274628686178930334115439194604975904573675930552057826257352113995200202
2
21/22:
0.09885979341236608235784167847059215902260146914454465468800410596905495970785422649147930918532343432
2
1/23:
0.4085468431908661913763831688314586189493801546722668243949774236995996256425997891981471316307214132
2
2/23:
0.6279508392626571086378786399920417245583205400222995435136843693977368928221192287799084916440090173
2
3/23:
0.7763091835627075421758732190230937290100638857917495017894315406600779469001214844201835347603460591
2
4/23:
0.8788931626631362437760465789385673464783693584426707345861125374737107694643703251830627510299599455
2
5/23:
0.9478183429444903253235401593630067062234628809907985563639782584943750278247626867215633783059208381
2
6/23:
0.9903105752418697387600310748080595148856597677659990314831582994868310169926636566896707782116672156
2
7/23:
1.011219960752009497831813973221064076190258933499930329349824541671029687421454068243977783850798331
2
8/23:
1.014059395001717494107655871471919886165983346034624807878254852106780144049626069993150343078655348
2
9/23:
1.001514232119959310093150232737926446924679955364248787792516889900362854560902259124091968582225583
2
10/23:
0.9757224794479959827736524312524423860858700627686948418026719627075002897007663329555920260818144287
2
11/23:
0.9384419279028603132408633220084627609698311286990131116839380354650944831249986643414556537338984025
2
12/23:
0.8911562161230914947121413975907438826347211863684545853648704159647288935092285769373592108295399502
2
13/23:
0.8351455314771096463757235904429104003022530618143198487787457672491690704988715049179179532545582031
2
14/23:
0.7715356805097131431247932608695336903344160305610232200611957513300504130737124990022082859497187711
2
15/23:
0.7013333347531410309729472000787712706544578869249154482186415191037779036073271606217135226406858747
2
16/23:
0.6254521204971529262694173427862972310230509182194187193192266660060044808845978187428736408759389457
2
17/23:
0.5447324671803239914039603760126875735682991745817717388007230915044665702380493682209911727968972405
2
18/23:
0.4599571032204923339367139437367855131805278496064511354626422771406248829743795202437673652650136238
2
19/23:
0.3718634651622774967222186432026074033953776854253583306469851114203206974395572901864875794906322717
2
20/23:
0.2811538959417726727958576816190639715672340019087499860478523909166624384037896560753481456545124513
2
21/23:
0.1885042579310889867498553505227580513191358608009641762206281006608815080899340661883771161290290446
2
22/23:
0.09457142355953763705744384883543775667021988466111705263813523900073117923154017480819288580871690450
2
1/24:
0.3970923303204925912816374410234241506270853041838463136356514687794856823918696047460555267840095062
2
5/24:
0.9358335870151282156920316221595844024802028872316932784450190991002448895889484529544773760120258163
2
7/24:
1.007125292657612295952591586763169076843883046868477173998461187744347957557318566568239337881492617
2
11/24:
0.9568549221901484349028564935406737133274526428899019831162368219685901742451659082016038906578699609
2
13/24:
0.8664966328758107022698929600117675926002547509641082243206777571243833402656974822057836971677126032
2
17/24:
0.6023729021073716774377116721497388917662450305252688688651461854773654779431412412372018557011162334
2
19/24:
0.4418541145026211128985429480329143834782288902288500212683808127365059156115355840937432054832296250
2
23/24:
0.09063925824268167124241482869696316023284026815713012879458298149144766951688513245347985066690025402
3
1:
-0.9015426773696957140498036211335874930737397192553741613442036665063786543397348176398419052070014436
comment: $-3\zeta(3)/4$.
3
1/2:
-0.1126928346712119642562254526416984366342174649069217701680254583132973317924668522049802381508751805
comment: $-3\zeta(3)/32$.
3
1/3:
0.4006856343865314284665793871704833302549954307801662939307571851139460685954376967288186245364450860
comment: $\zeta(3)/3$.
3
2/3:
-0.5342475125153752379554391828939777736733272410402217252410095801519280914605835956384248327152601147
comment: $-4\zeta(3)/9$.
3
1/4:
0.6635907140443085628718337198451323610429572626121042485940424976293627546659629855142605546437471276
3
3/4:
-0.6917639227121115539358900830055569702015116288388346911360488622076870876140796985655056141814659227
3
1/5:
0.8136972072801448618411803234835130126062862075037318255777688995153616363191757112170024133420768481
3
2/5:
0.1893679150108316838794448047664136136312742297442338898539828423610234380333963423420028752076767393
3
3/5:
-0.3809567221426909190972745853393910159308911422611522281325511395922998822361029987498782988427161551
3
4/5:
-0.7663552285274369408713191222919096091984676500676733531142731889251057768108266256315016945401576632
3
1/6:
0.9074855118950023857452406882452452067012437435599127238476716353855180522825850775820723483278895729
3
5/6:
-0.8073141032983695286285958414526243741374948858648711503649823391070315351337256533998676921937783014
3
1/7:
0.9701635023286460649976267938583627944140422403505488838082272935153742373527538874141419450913596159
3
2/7:
0.5520152587049939942290433144814274535254959008674236627833537690975798458128559671034167362066577361
3
3/7:
0.1005782139523595433131514880619771780685314065150415988268374464382765047929517947897395036136519465
3
4/7:
-0.3086181358654589349232429793714786608266984727984648408549346592485213237678367686993114438735757419
3
5/7:
-0.6291697926713587279598540593242971187321705190670398729970874339097101767548151592782445971750516716
3
6/7:
-0.8321596876523975664403659652380059310326682651336929681123888512409792758995398956382877610396951819
3
1/8:
1.014242716621058948554852043594917589436183244432634891677607527379054509498988792802572123621057704
3
3/8:
0.2681421167725466749959208615063634109006062450955268072623581110901585348453187650410641421363616575
3
5/8:
-0.4410830974505745634798933822577526534509841523052354800463703266420803067488386896824405456817281382
3
7/8:
-0.8483450381099818078368936136336344991754439287796088295290969029717138208324980464240069849601209218
3
1/9:
1.046498461827414254325961121098275723469596674767499129263923929902342583448404067417615777954259372
3
2/9:
0.7481190287895360175836706540089160411408737389030617318604681877558527790824050612398926364646992688
3
4/9:
0.05198884111656365035132400548779980830202369657864732830044874185731237721088845246348055651613416236
3
5/9:
-0.2668560172740712128338348698868114865981182565890807523751504625981572243803391529903912112256553011
3
7/9:
-0.7351218185103951049958396526369660890653678147583998997853560022915246847796829481240224973356657282
3
8/9:
-0.8594687046300302499300434575960467131843782400417336962988068829633793886778028021076426188380845550
3
1/10:
1.070863172172536476096023280830535587063360450310017445650200060638633801525428386708515213893431741
3
3/10:
0.5067991141647550666792290793348497491931013587380120279018677660372055350951489540083368069453621856
3
7/10:
-0.6020382947004277964535477256696975031758241443033000849350055509352805056541747036958063816560412244
3
9/10:
-0.8674388703525002606357281999596573339117888984340844892557578357597933924456344589042646105579125291
3
1/11:
1.089750104591046335493797928230215878132944983775620465097471782620289347588482702263685670960306755
3
2/11:
0.8656279578012325419888803143112035598900971037127757612866689387569197745150330869011133479747797850
3
3/11:
0.5928750542183976788568988981755458086879495147187362528241968886638283566709042817248531743645257463
3
4/11:
0.3041846387191008069299945399005907223287548953190282337059486821800488259660856702516934734786695183
3
5/11:
0.02138878237898438224480569007040601279941530232920135367905945227939762153296529463199291893386663108
3
6/11:
-0.2397245611641689322439501524835097598395204792910579848730105892621624725228964022665947524255195834
3
7/11:
-0.4673861873464607849094661489746965344907123039544034209665610661748910696664927658245689390963959288
3
8/11:
-0.6528061945094399119178864362964232486478296345415007490424495359793689748016283822915018624709056421
3
9/11:
-0.7895817981214573402563816793360558793079083798830187028601817682119075680235116693381899796051124054
3
10/11:
-0.8733431151407381999965778496524149881604207078474265247758045479310594039597244305384073339666118091
3
1/12:
1.104709265150670330080297687595296987758723294114671897641645295190412417890329201986175327699746086
3
5/12:
0.1373845150534876647792771197993469950180649146304967246261239040249271404188222027913405066205303511
3
7/12:
-0.3392130408780800469364260801625030885524386360967145122173694888016850242022536161413074296689749264
3
11/12:
-0.8778378871769197336439875155339856860834123582246937166797273863440329048196829325906572406177736924
3
1/13:
1.116774612025198540885209645551239864262112475138568344697408316218939923584784787912534927376441946
3
2/13:
0.9417451200131862635598025553848501402324091478588353510161319275472445564437451424714811368791472521
3
3/13:
0.7223774130413802554375197260696231364538577363950245677025550494504754059172023396814443791221178085
3
4/13:
0.4823656420555857776424398188587803457965059908895233103017169151940734950379471186899145455129396067
3
5/13:
0.2377480676748856523028059042462996730699286747308987456615185925733070620661022820322112260273061234
3
6/13:
0.0003621590249435583557826780224144247823166306645412895311335599726545058497802653006885692834713935538
3
7/13:
-0.2206967420304190521045974296986712570833191777080061662669773935556867019682423908743547300726351767
3
8/13:
-0.4183197859961717437649494183873045556344983194131210569914615074376350069297916003712792852276186134
3
9/13:
-0.5869455885546287135836771734556064847051305707428035745495822920534822467703950187827343668198442600
3
10/13:
-0.7222868732851443658485740565640195302582785787288892453197716594573117794547572733562722368012499601
3
11/13:
-0.8211537094992898191491926006701550537832826501364545234407026987487261826842583627990025005009123505
3
12/13:
-0.8813383320219019749952590067050273292040101881738595069433753343321287844738485022946646431566551326
3
1/14:
1.126658231775247596838715430364664797741877328723380578086554565297339825088564198796093903155343283
3
3/14:
0.7717772964277444356567227025672837920127442498634222892218925886856162676991109362417529560889729224
3
5/14:
0.3248273141887617089210371384658263685521638796587464276309802849494277667304878578546034165145668853
3
9/14:
-0.4821197623566013909110006532969006482352065094255063958802521434268553109191916476741645658083298032
3
11/14:
-0.7466327429396545498284348305517894974956113982346618895151832270760471415008729875443180801855599357
3
13/14:
-0.8841173561930860805893087319000740991383667686357433571344977419184962657503757269425584168825033786
3
1/15:
1.134864088585040323321624142750000934847810634778557284778223908671575369823768544931014914441680353
3
2/15:
0.9940150892999074174812266109171209084045363251672988372319452229237166092978882080417359742338165536
3
4/15:
0.6118418203017387916417261161107615819822392515127572019368910729212776931535860810639667462821986240
3
7/15:
-0.01497109773819335550341575107623705702296426657697253640032990477666443722480657885791126999466730104
3
8/15:
-0.2066189813268225117344136214289431199137048717947106074672294959584971171500175443722339602261392527
3
11/15:
-0.6634965656334444195753295214679973619606654694614348538036984469109019724410904671570252363387334372
3
13/15:
-0.8410546342244727195707950818894305129089765122891095367477224546933971860094916877757442876632668976
3
14/15:
-0.8863603162600634689513174900207207077466765534867325754702376029406462174992964929205809208832262148
3
1/16:
1.141757348339246035905854095882752896424045366592267507699569637757275889022231405580032333707181647
3
3/16:
0.8495861456295102157009631366591126108427772956164090041313416068501703501900810718837395419663012864
3
5/16:
0.4670721573087007947459044537849909117840475592507239054278951774048383860818108889640742097399395156
3
7/16:
0.07317424814197997134485809297822542831647102322879237761448349997346083765539440558181023308892138493
3
9/16:
-0.2852605076694754233040814963866340531103320054236945849967577257163892928635189171878119793289516154
3
11/16:
-0.5773429316713444356158777993494290751467935973270327754394877590653584627690205613846843461603715501
3
13/16:
-0.7825506164363735469519829212825217581176257343425273023157520790776307164787513806234735064322108720
3
15/16:
-0.8881966691839812987671410849840234990649995554841087847801677559125122616474842073793893028019172206
3
1/17:
1.147608094677100711630088386597023697835565701963006435736177115967553596919349064046222080034565223
3
2/17:
1.031556219187326711841383802006285428898657705674207815366677263144538210420455926408156187609383557
3
3/17:
0.8805593187206133504133338793333554013099269389324312954531781571517747490777414344703831499755844837
3
4/17:
0.7086475576663593645916569504887345581883106400093661039555915850508382318195885431393613416966306221
3
5/17:
0.5254462096941504544132911541017924384341532758428067428805288053968303178070493677555865406221379519
3
6/17:
0.3381985280594638217068121172003103473501100441904327939603418172324121646307389697546866280081583397
3
7/17:
0.1526219876113843903330465725687222166401089188654589747491287219994217911345232280372277681243056468
3
8/17:
-0.02664562537954410323634456189929734399706422314876217841467181867148856957806668758728769389307167932
3
9/17:
-0.1957841345905231551810910471245657411556610957373514420589551313738661915004920830237580643899831541
3
10/17:
-0.3516244092878212390797042850770416702582087758366647489899018977103397181145374010451556413676918715
3
11/17:
-0.4915366060634902627526164235430735613497864116409260264923428871885159764646598992589054907491735697
3
12/17:
-0.6133523120161057641832172253710528559987054698019729301280042798244152473356837180168754509640609198
3
13/17:
-0.7153089640112453904007430909635588941875766957965566485592595397187103742141052150361832651698985419
3
14/17:
-0.7960096867057473949866308500332778144723994278848230969630927028436717079200566920317114929735448988
3
15/17:
-0.8543943297707368706934695643841017893515800456718662893777793668818286524655587906233158521852259017
3
16/17:
-0.8897190398802690336697424360954523406109012755444544818945078001814190443142350824441830331322193337
3
1/18:
1.152619728265084477496906873460504245100183351523784831294330067301185550279259692357748880364959340
3
5/18:
0.5770177621328286515058754749124664596225222203323295542963888311720686375574112652546574403564305548
3
7/18:
0.2242757836422500925619193581120661463727625307368170335302955493083055407337778932042577519428774639
3
11/18:
-0.4080562382698488688108792712713076686391044844264170084766345498811867119286986302352633762767938960
3
13/18:
-0.6437317664513464547143341923841693312720517844795997423901764468216079436524960535022552431628443801
3
17/18:
-0.8909951128082309139154165931859353142327841828319100489783490848255999044171586755033449358763944967
3
1/19:
1.156947904006082877994539100033630310669074297717139031580111394234482184927526599097772103028341601
3
2/19:
1.059490515492501894998914124022126021735730886369606441776050578745625512396020096501765786225091117
3
3/19:
0.9310492892587727918019955426772996053835437335785841073531484797540699665994115784006828415091292354
3
4/19:
0.7828952057740676206882290252485240615516902182864256441608027450952608960389788170696777954132675406
3
5/19:
0.6227888648050821327865856362690860105379553304215799365347310570528914469156640785727238119465360362
3
6/19:
0.4566002534671740466487561836032134982854395135372240033817553638890545273350788141016449475927338262
3
7/19:
0.2889946793031965992791705846687639374367147810104900077536473630512691497034431969973661217246429282
3
8/19:
0.1237904794979451086712424652369019768414098784278520408321764922667173824364200991639415402456309121
3
9/19:
-0.03583044109244994853814352498439108940057629782156475034058521671747847243536644333072607581285696127
3
10/19:
-0.1871883776907894025230591172726336119082090962096196049434394706695404792717639504123565883051602441
3
11/19:
-0.3280152859709399287061940894310260345444558422264215674591039019621689661278305678827594413983673568
3
12/19:
-0.4563911691101414701335081885655750408154666821289862171962950942313382913868444635413193614803494525
3
13/19:
-0.5706980457447094141821332307446072584893061840694705576808291374468891001817899299869747879628211894
3
14/19:
-0.6695859592277794834173504155872444135150076044739848377705909247202765667336050661758129590228260972
3
15/19:
-0.7519475858995813435204184089392985673413377486794626339527586220285815504298737922786924103518598126
3
16/19:
-0.8168992258919792801398064967764962308121838551942781065077096387818063347656418748752716046109457789
3
17/19:
-0.8637667140489849898268568677099950063478083317980000307358498924718102883862753922343463373717742317
3
18/19:
-0.8920752751329574042448105690280988052351415761247374211360987495480758068285215749723306564720688216
3
1/20:
1.160713454180941891072765181340472671648906290500340981476852273450261987148872898232491880821989651
3
3/20:
0.9517958078689263013124582550856354882177332576509564311945804040988192196434167928146827667160005923
3
7/20:
0.3475638278215754256254138344605146155011054038512135721122711654372655104375379429029517547007525894
3
9/20:
0.03512629859910411439523540125029802841267614295821773170980058581485193608393693116323096582413361320
3
11/20:
-0.2519860161872291795541674512402123618906233675667388540237400447548002841953455458892971184636117455
3
13/20:
-0.4980734014966823747388007658779389912950614399270385933460225531710856368510816188269033501147628955
3
17/20:
-0.8250960293277375346426509852519230509194579179664534242191134625895178358696295543125985649796600459
3
19/20:
-0.8929976611378077720487593611328387748830661779228366200643022582906035367675158015553630773486317159
3
1/21:
1.164011391195844032980746358687229994301810697582365225525794075590127706370130228212003077245283438
3
2/21:
1.080879322568135057435610274122846221606874747565537412613285298744438648306322124152643817082788893
3
4/21:
0.8411135477754288020301600714080438016820788512328716638049452105028167275804765712167884732152347375
3
5/21:
0.7001149669306562327304905393950473021475824594570512452799796125793790530899383910833466843681002987
3
8/21:
0.2493141143318832114549820350930903263709663025864802131321868872241071678251892972217267466524067921
3
10/21:
-0.04324476410808222835904332610020216629682152752123034467643798806722870987736298321175753927260352697
3
11/21:
-0.1802031260303703387964176214389274434282014751515148734166801996587758011960244410817029914710430268
3
13/21:
-0.4270456538213024089100448778231147873014132629208199209944591646231537254650090814432982639489695873
3
16/21:
-0.7109261579576767898202513709200978437217878413373588314490891095961862305592791368862860691862511805
3
17/21:
-0.7787850191924579991664145626347712200893372755862516105218984886967899356241792469113567865521330395
3
19/21:
-0.8706009356242778569280702562708352711863550347573194966620489961187344664112029813484466987789802087
3
20/21:
-0.8937915605538102686218437901565184389000920106909808723724727509040180442935496971738421229745862152
3
1/22:
1.166917296334206890950858429820387569336913628952188103118229969619322978916060419147837392087638752
3
3/22:
0.9865082107180356257295733847209323314438071153228620989636825723753194271494527440783129805166771504
3
5/22:
0.7329394827661536041973452178173697216009347383482402787117164902356147361646196924110844367950785517
3
7/22:
0.4489809576456093861728760667925663912947823046461671617790639165290640723603044007888171553139744902
3
9/22:
0.1609521524336604665532214265364832561251732650363981848935526543876860113054609124419160008586322543
3
13/22:
-0.3583476019640248016173168463704972259521503600071528606085980964406629033113388297764634957599103556
3
15/22:
-0.5658275044822245824002426040362405249174603806347680170207041830727868397769275922449593900880734724
3
17/22:
-0.7275922871714075086361437952997682184010809127659399402919516271657653307813783687530862070616118940
3
19/22:
-0.8382894471634362060153486601770458792718197366431780357576333502093623379817266736470996869255457138
3
21/22:
-0.8944797701864453070774089477628335998036773830082829868438620239642506420189397435819159743475620629
3
1/23:
1.169491929802185540821037025680269420775792451982057423632712927162576654107809778026665626092130509
3
2/23:
1.097646730177390592148957703762824482773409850240442514630521033174063213526098544749133504129768175
3
3/23:
1.001121189907873709346005841578089340900980925733320098538012198726692065641320517138667949062175337
3
4/23:
0.8876407780274183084104694035243035654569695434958374332985933353351363348542024511981838946214273779
3
5/23:
0.7625492040841999580586210852957903243901593867210206353138146257167800306893259459363073387002928114
3
6/23:
0.6299140053594477495698102239069676355768037525319990081877113221867372177921992806915418189090900938
3
7/23:
0.4929949377972522562580318344155332371850737953553377175949316227895691146449061995080237299175553353
3
8/23:
0.3544884568161839050917687523759150751725180857490676783779906016228579827660625650781433276697254557
3
9/23:
0.2166718486842001456853971215193866820371179972761194726950755473646793760034318569020506513048728431
3
10/23:
0.08149538095603286445940012909865204599476235864133398699929207102878810643241331484468409905315569167
3
11/23:
-0.04935526150467331726569435922759278963621678338469569135928731226424999989741357172814153140962721326
3
12/23:
-0.1744148003097861559302550407072980353843932139707910073844833549530152127841577137317040311052949030
3
13/23:
-0.2924060530980358074477884504989889039964573555414140736220631630737900467307309890861297226368813951
3
14/23:
-0.4022156883954980811713398847746685102599851965412912573360734493545751270237375112083347297891238152
3
15/23:
-0.5028756717902155992294854537782151663600356093717328113602280979049112068662727094056701807609345280
3
16/23:
-0.5935488598961267765508668056092003647500700944906605319289499851282128964008405773101074123648362891
3
17/23:
-0.6735177054368942885523739840837921444229020560246967600338321609249910209882387091244678266854138195
3
18/23:
-0.7421753588451917419437710530211273128914687970606871385639916079595830040812226172251363139370038885
3
19/23:
-0.7990186638233723321375272154303247966638400220585705137040956849294218391626868099286480627039960140
3
20/23:
-0.8436426885680117719535532856013474320067799876003203464910843681800077611932706969657824943349028138
3
21/23:
-0.8757365356705360150463403528817485914091674643664831563058726993402791298125479319495875304744113307
3
22/23:
-0.8950802472578378927837975997395633000824399894219467949750826688690608507262851418393822500596884650
3
1/24:
1.171784608006770097678584506411383821217953086051527600459189380571908553672244048653649084332481833
3
5/24:
0.7893529964255315121275806819408643918051655291468752844594245439675756841436376672924276175438771844
3
7/24:
0.5332046151565578615140454881580145523581722475629318377276314009095120453729543112911514452756146624
3
11/24:
0.009982808634386455322658111441908701431280160487707380620075656043906991494544969166863751229385055407
3
13/24:
-0.2294422804286163887336549903254051229521332500438808097900075026299152176994657023145280613838284785
3
17/24:
-0.6180078753760778732481520081986403244962819065871104657819737731099333014235177153264783026928583940
3
19/24:
-0.7550068676621595959327614019910276430506493004892511033028935679613438990389321165945924908887445966
3
23/24:
-0.8956072917191025151585100845125595742782722625228596260487780567743054491996617481571052524075453113
4
1/2:
0.9889445517411053361084226332283778213158608870627339107819924016390151946980181964119104689997999338
comment: $L(4,\chi_4(3,\cdot))$, Dirichlet's $\beta(4)$.
4
1/3:
0.9158468848305221005784557354615554066325218024536092822321305122909406032199004637321743668969060628
4
2/3:
0.8140861198493529782919606537436048058955749355143193619841160109252805361954670788730438816861387225
comment: $(\sqrt3/2)L(4,\chi_3(2,\cdot))$.
4
1/4:
0.7763467588132879772993162190712448513663360780568334458125130664503083458099341888984933158365151334
4
3/4:
0.6527286898456498102857633899176976237018534671739917069647640162454314464726819143470045072115401417
4
1/5:
0.6602016171312831515738235641315914321170768954262316921084986545434146561830218220717369697003980130
4
2/5:
0.9775723874383045294570230994546467054192907556417957254265457136020612609629586513857628653125945892
4
3/5:
0.9103217538506176437830611398671766311768325617704198221227731585591666361426271525546982888705541281
4
4/5:
0.5380050687014950853916956766997605939396655509710072264301804403431569985626519906485166115363236894
4
1/6:
0.5700264941676659537127429969921727498931271466858792746190597726828583331002161771227700300050269650
4
5/6:
0.4555456335638506911404360300594783240640619213791781143400434586464907576977286191562482341429137071
4
1/7:
0.4997770699521202889573139708351899288149300730931968672365734982297672158137263622442845549743480482
4
2/7:
0.8445659560904090706670097183228998455650724974740546328990508272945172397816007765786245484861235923
4
3/7:
0.9905735811901512569944199497255571172034420656432975723599963669671021526859917903791066410581198053
4
4/7:
0.9412977905100488626039277302199536861885300642796800675944723488648580825808635072326120802564219804
4
5/7:
0.7269037322766529628415187520454056347915062394390946244497417836864099794589928381745480384540708448
4
6/7:
0.3942063254408191551239377560448274481192960109089400381241921448179525608410262651719564864135825992
4
1/8:
0.4441017529328836565051136719735023700735649458430045645961088169302214047713574216944898441499230724
4
3/8:
0.9596114221556715993302722900544308629637549859510133285936626336408818543401161949568731105312313839
4
5/8:
0.8780203359249653730445518663147186600010233025542643652230671316102029235310309556634975471297888661
4
7/8:
0.3470584080812226593426991445895967636527729360859003838695446836239328615451156480821781796703586807
4
1/9:
0.3991364461961667999981783615848142516348589735753307582114626184270784612743102541566402099533798809
4
2/9:
0.7147314619025452612861125967097769652278542957175973564136315627283339195614984001400615188120729490
4
4/9:
0.9943751375886956451682690817153652145093953736054758761110155687430639543648633143209331544070496845
4
5/9:
0.9556507609064020648760922958408536981367232252791469900285457346073093641987229450535415893943158392
4
7/9:
0.5904345697039583056400789614953563134141798740169128718997546166354509252658904858499448745111917384
4
8/9:
0.3097950134583486423374142869960921309813771866106310886597586730860367213291229541391325201018707623
4
1/10:
0.3621775847317275356141553829390375248256523485606546567512989816435291700470430464331640705672557101
4
3/10:
0.8683265909402741880870103575037869991395540776902824346250593693574652053647354050086567456392912344
4
7/10:
0.7545363717089469826141277150203899202424500074689799568597127245375693758469070109393194595304719684
4
9/10:
0.2796523825903171416674274374225885958110177366323756952377366498256023380241653186741969493547059585
4
1/11:
0.3313217535242128624183061283990080830632529260314607886953454774502034728661106846988585745315753581
4
2/11:
0.6122320259380662120448792277748060873482709093130327884932148345861472357906519957446030202781873321
4
3/11:
0.8212093238049807325366369686638182544946193139073176670767871748515707292250911840457026566551910647
4
4/11:
0.9493957618037376464358020113063001218717824341561673273465038863365581166373443394763770984666788910
4
5/11:
0.9955386992739550956828977813607371299997826742614788992558416608010951629711841116378827555003133264
4
6/11:
0.9636896054887107724438107532448424647316927914655624379891299579102282794221492134840348558757130837
4
7/11:
0.8618022464138761606881569332427735035713877197844020320554856450724590923501790229313523110078380248
4
8/11:
0.7007481231188918859811606245082129464031577149741223623281459301127921942973225323601982996707269292
4
9/11:
0.4935575557125990062404039763615185721142981050435118725749018487940774712109839533100558829698524708
4
10/11:
0.2547927502819545859126962249271573221447190623673316901336936231269350683922791852307831969968019416
4
1/12:
0.3052062887094203867166199245051782926743452727971898637143298961006792894432461464708990507883490548
4
5/12:
0.9861245990908001268282964767332275642410088476229553740155803422157995493018502793643422422367713679
4
7/12:
0.9291813948953187904357419729757927737330011074505581097230749098849882045896342019698112129689071545
4
11/12:
0.2339529769384621425025270498811566989377043794614549543869474245153219978057191243305527970377206842
4
1/13:
0.2828367664763869408985603671058116879163080420364205964397898042409431055338111231565177146451256602
4
2/13:
0.5327821262607975430767454056184008824459671865175458390777201431569357088805163793548415993529005138
4
3/13:
0.7344743563325139149135935887583490664927913897760788059062243241569574227874710266100108971796651450
4
4/13:
0.8802788165381929042876905313812564244491535326450034572091159559834911365370298432500963448781140540
4
5/13:
0.9672520240172205410584747654928909590705279630102230106925909169179092563167051580414374665112104019
4
6/13:
0.9957773804666074022762647460574080592736426307416224152211291986770771277633532436596232107638200006
4
7/13:
0.9687475053798839962745188471319691120723223627764064944017448503837863600228849216924778964230683636
4
8/13:
0.8910017510454470423548422035552443266850797053935510101921430201276189085895736741473677170319369463
4
9/13:
0.7689035976575120239933352559368508836135185694708095809350980784675387729633331339816753802491219357
4
10/13:
0.6100021837741879896290604955011730590835860609333760040035831743223227818170518711525579958341876450
4
11/13:
0.4227472741935234300407840891957438293898229949369204069265806486589993168133876489485795562431362571
4
12/13:
0.2162390006937872480139671914035115776105621437217273665550747863463261419237465757371625147260130960
4
1/14:
0.2634736352693224384950127960894165585649941895728132015016663689988217289685431283169588411975691133
4
3/14:
0.6957829524715199806489975357386914032067275793589426877343488818581538553517751534631234856052836945
4
5/14:
0.9429799045213974745862538117769486696011860479274503019762995823086948369182527208767046612354226725
4
9/14:
0.8521169379868158542310639677712729652522477679975634739200818593478935894858786161048861564286638169
4
11/14:
0.5719612548227510735246950420229967635562973211535304911893493359872660862660261796657351554730187189
4
13/14:
0.2010015015253074023753485497350178174631279304361635930970946817201008269918273330364232718257756073
4
1/15:
0.2465570609077386177889615646595882287219425071423786248831019839684072669975390191778559102365487514
4
2/15:
0.4703911452028732254222780074328998170492679130580605806736673598091867936415368352369200308995672269
4
4/15:
0.8097403333365359939344276070834213461966424987787490148472884622151044542284877362661707223287259519
4
7/15:
0.9956596001541035390956662266099088908481781049602664302920781729253738880036781661066931310914465653
4
8/15:
0.9721898291844311060192691248935994218993301027090012987547164774263415232796348017600380177946837093
4
11/15:
0.6882166046884821056820189664717214184592262359401238525029489025368117638185333860461659701043904883
4
13/15:
0.3691736035358062261804745565474721487746876007107169538177563020322987368629758682036486906084764829
4
14/15:
0.1877581677573794646111768137304757515907840180101210522988935639922589177923469147732409063741028480
4
1/16:
0.2316563099707799857807152782174268555058332403996766240282952428970008023845576941686865904353998434
4
3/16:
0.6275405407612507598425961430121576528471085029297420199989244895098905349532808797482568310133626151
4
5/16:
0.8874488999523385505839599524020760394561048155662606635445823891518974602365160566555099443069493721
4
7/16:
0.9930100317963805954613402699378624981966358253596562621733566308556578836659529335456577432510567574
4
9/16:
0.9496277307862277630435028768641629027400392083489187141896635454026662759728134775353854707922619223
4
11/16:
0.7776963579617178789533909691127362069559769027469776178916989977006220947951371871975727509157257638
4
13/16:
0.5075891129917918099263121067553537949766391296858653539247166603047803031607663553786476921969586921
4
15/16:
0.1761435908541695287175760692207390592466376221693010534537816407807231267881380164568753599166594593
4
1/17:
0.2184350423991999264608145352599205100272427234449733965509163671475604686188763729958303033684494153
4
2/17:
0.4204718333651491222735385609147219480260838998534735931343868524943185596808465047162773135384576874
4
3/17:
0.5975638547418873162471514224436342736056039835633773345816945485175204853378010101468511118832922976
4
4/17:
0.7446456977478308625148917958410866551433576425649212223711165082806300445339820605046392737640426649
4
5/17:
0.8587886108817834296922329000223011735257495189434280904007796738493460718663568258373405600136802652
4
6/17:
0.9386039550858419168810491604453991204555584224904185205001043665352889004162304842547438175155420948
4
7/17:
0.9838916126100050036177445184407327848914346133505881197417371213209868739197613160565577865176334985
4
8/17:
0.9954031958040798494822281041646766601353364683007653827121830116334773881868834233982447481782828385
4
9/17:
0.9746686879005103137101503272714791268198386888053542330360432603102560571045371034099577932812650629
4
10/17:
0.9238618175654853690381169837672378619478522157565787423018394334834107023215397911046817696809023065
4
11/17:
0.8456907196250796970607406258139745653577749234936865520484731484522989336567099402610869001650961602
4
12/17:
0.7433058836860977585624682770513964407822679919738557476130497446639197340761643519492553388035674769
4
13/17:
0.6202202982723208813296132828205020726264405840273255495320936318264453710106216325798586802417573100
4
14/17:
0.4802383603561570766370202773879593835486591807520750195191815027006093727857721996150081346938495358
4
15/17:
0.3273911211466702644591770864345861161331641945328584403379972889592398041140987471531974043179523543
4
16/17:
0.1658760632285562861766222151455802665239822359632891974091180105857706486587705599062956391761422044
4
1/18:
0.2066270789188173662312656766313040841899578704752946580008220769505040535905848796206867689510421314
4
5/18:
0.8304905105954532658701467406618280899590153504894896382407688294446754303052590153326219206414101772
4
7/18:
0.9703534732373849416890313802409756399515024447723428361192305435863690926181207198816453954170607973
4
11/18:
0.8965491520243901534840215100540561007747299605202287271317612165069377269598844091504022861031618300
4
13/18:
0.7110341654821530077606352036817213776919249473295962644872006126187617597804186472009292219671206973
4
17/18:
0.1567350231442965162314933814332023027356004987783783132243892496471192459312960978511067427068696463
4
1/19:
0.1960191855456121315675597736870554114067143450662730625100131984452610707299333968259815142211262066
4
2/19:
0.3797903903983936924309864264127958897141229757006068905794085950940736966973870885149676404204462487
4
3/19:
0.5446922851394406058245979591143371440795874689383631382375923463915461694236866867969814141833973796
4
4/19:
0.6866021655971105043702732092879295010958870797900330131172274351149034639420328393760784122235530674
4
5/19:
0.8029355610205872110526569217858421028956122181437052322582062227756162460957166590102166995443519165
4
6/19:
0.8922216063417439452995068534714457130042342330403040393047427166550409651278273651757497394144059272
4
7/19:
0.9538539384883078362866112562674861691039167332878078663688588295311530516037764016694423583485355676
4
8/19:
0.9879239092534351401231619463468893708720439755907916349265142527454662323660885685419805678336197533
4
9/19:
0.9950993424721698524247148032010207332442137861887176889777976707442768266637825936979995435176522448
4
10/19:
0.9765311691289432374230039944028387425951576645507555388293492802369640943334375923527982236215809416
4
11/19:
0.9337783362100948135403919960430885671277867018647509932609518143942579762682500352732289807390451734
4
12/19:
0.8687452928783741729897635784568002615950145445284502676307916341879035848157717816736777431495787802
4
13/19:
0.7836284447319471736757864061643456803048574149742477558508937623815530170258558924665400215207085797
4
14/19:
0.6808691648794693063747814224854872600712175100748607899045375627459957343040369599661169215916542988
4
15/19:
0.5631116769404311118548779659945683297368815828411840587514131535217201848962717683083308412443505983
4
16/19:
0.4331645843467226126621596024304064299540581898083251333244995068096660487827082661500126967565966387
4
17/19:
0.2939651196987548793847022752518047020771370907268527639397551657047107637046329835929578388925021153
4
18/19:
0.1485453867458129200136864703854559251924489731036972011875871240585018586427600107616105591685704255
4
1/20:
0.1864386417947796882195169622136211353162965830605326432861308754431933003838611067323715481406326309
4
3/20:
0.5213421425310207882625504774683150654247066585877514144014306356201368627672464186221270812751646863
4
7/20:
0.9354357435508819936256991427440356141361897268099083937967892645722320325011938961506884133231234391
4
9/20:
0.9951352470087719047586659174203403708004149006713560149686242954556189332598396842804643308504697889
4
11/20:
0.9601786991849822620502374877425167963240376835923090530639072142274186410068190194461897121811315441
4
13/20:
0.8411186970872636207989331783664868741058834758762858991893251740050358605203305197832734808818144430
4
17/20:
0.4128013186634865147516741827803416905322623988764661100732982144504537120966544929960449880702532820
4
19/20:
0.1411664437033137462677475393462414447130900394904508111922185027377521541279807259282260393197256672
4
1/21:
0.1777441257731677481254083482934108823001986424967617111689466990809520432118163104624155624899769646
4
2/21:
0.3460822576276171018181039198155334919041917741385122160394497953139680901452082866255304448069792022
4
4/21:
0.6354445798605953496875931739054611595456519153893505294985099064501765754323516804493646951982021477
4
5/21:
0.7508442564997994243628769822795175900495496046113836030430359315774213188942346059643596734640296065
4
8/21:
0.9644495653845387864309351597488198766320880566883563897251332595605583620644873006509244910636139863
4
10/21:
0.9947880241714126950854595325347347674357431091027468614154993716805417998814941120201133267874728217
4
11/21:
0.9779775437251982437856913627451761179592212752615659008949974373472035458885359526908352946763347512
4
13/21:
0.8861375961997421817462858421597354086170778421916638590543709457958891628258563204056938026116645483
4
16/21:
0.6264957534783728374771945407126757441200817159735402453660985101173535939090478419618455076155955038
4
17/21:
0.5148883841875280013837262789368586749666409083033059807828682490051067801742907678679991338152503994
4
19/21:
0.2666516851450426831071547730773508469609852847148433998521360570076960182161643265693598579072039337
4
20/21:
0.1344838435697156103981453583164691958121746707294476841640154746667060319436652746342242568891045643
4
1/22:
0.1698189363615639358195850859752915617732774485105776139298521834257522823975746737728536124736771137
4
3/22:
0.4798184763579683112717534623090268524727738994641434806219661325964957412272061248751040656134896225
4
5/22:
0.7264812880868769964256868816231292877280270578379598964957305327309976708279334195485212874186931752
4
7/22:
0.8956246788725916150731340832211659564395666808874264024676337417260430350692469297582853929315133246
4
9/22:
0.9825746133854130759097128258971796665873597691805131512519247628945575036056301731479811667578023546
4
13/22:
0.9208799189213382001296623288519898450730725060500741671800620317952978197042571789842241813865707958
4
15/22:
0.7878993980708570949871144665658192684931432159143761484606980360919856485254745518918663540555335715
4
17/22:
0.6020389506776326094653246589530371464780542235552750340887503251308607754565354055937859429811540094
4
19/22:
0.3771673108823457197046738412260495706609464852257287722373677357400494000740697268693912335315907394
4
21/22:
0.1284037171710373280172968199254155513903708327566450153429339987444768482893108381854962906572301939
4
1/23:
0.1625658871470270160066508167072871646555072215153598264412762753944767383034054054519412539919587582
4
2/23:
0.3177407000833720118612939962903542609848934506762613925299302701939870701649141339070385101737446337
4
3/23:
0.4613071683274457989378661483035964130849766532859698385393526199471550496727772259353439671012979623
4
4/23:
0.5904602933010945920111513280345017596515765488371192667316269211548065867603968797617132162374793221
4
5/23:
0.7032678114528623140186607190667918036593344494709061963989236785792440052817434631242287246568366625
4
6/23:
0.7984327141451206632464814374638897826736220408041812789865391213022208180931642897246102869105104953
4
7/23:
0.8751547527389295398861553131501676697933929592409151793670002650741563700527337335897479258333254474
4
8/23:
0.9330383943192094887303706838520663223001621936688908479023234891571630659585643461578041025789701397
4
9/23:
0.9720265090123354508525534731126135829388184808864497522904339623951708265729536336875399056690382991
4
10/23:
0.9923498415983986518766520044362013542792799606336464800638175743557467048709612151610817001914678746
4
11/23:
0.9944868663910598372792032227545974075854104231620298317849842423972752208273823640108581631523592855
4
12/23:
0.9791308289364841479637045893582232973313609806394264377381048684447466765107834716716629831248292006
4
13/23:
0.9471619567159854373726450083586250804979332233604655827305631706326995239445697539461107400367823558
4
14/23:
0.8996234988557026405357935707985802536420154240874872039915656521720738681763444697839032166524254018
4
15/23:
0.8377006673964636500029730442520231800391445647651434380438166693380821738922280117994590996099986056
4
16/23:
0.7627018153819667098191811168003451380881410312299792788680545585526471365306906748667600237517722722
4
17/23:
0.6760413605280601447510183637941118705072019182242529742692760127466274835293163557656524140199068453
4
18/23:
0.5792240812530624825340792185122666343744244543917003863909464817847756671728733112290935121329031782
4
19/23:
0.4738304940111934059198549925529934693640562746285079107438364850101612035155763364919877034151080547
4
20/23:
0.3615030790593057160320559686206101902507738981854471786660352297843774474111316897197676812374841504
4
21/23:
0.2439331634207351878599000802860415410284463820716214841884769050496362468198645239368243581440597184
4
22/23:
0.1228482996366055145239890671709928820323955401808271523750349916202283545327911387135614402202406790
4
1/24:
0.1559034131392369111880923590657003746264566788720728884261628851356832763442178944104093375840390151
4
5/24:
0.6811861850535274122126716035646133106826185129058392370905227403961467776114897383562169289616693751
4
7/24:
0.8547128102192315942351907232033443217235460193669558540158481960223735951594565002657771979872307666
4
11/24:
0.9957253214338254281092762345869982721131313266749150148406945672937758271739746103027100142938877274
4
13/24:
0.9664811993165176602964603533518536847459182792422331455423261392293605774482597197613909146641726419
4
17/24:
0.7385651358573167454307229765813702250069208809356360903004638322867500695857522250195507963661173723
4
19/24:
0.5579206101671773963591345439729598651524924069529698153385751976191718339487584534356741486820729541
4
23/24:
0.1177526270505593628485148685025530880421635197724241554618716481230983651638121261015469562354953832
Definition
This table gives the standard Clausen function $\mathrm{Cl}_s(\pi t)$: $\operatorname{Im}\mathrm{Li}_s(e^{i\pi t})$ for even $s$ and $\operatorname{Re}\mathrm{Li}_s(e^{i\pi t})$ for odd $s$ [1]. Here $s\geq2$ is an integer and $t$ is rational, with $0<t<1$ for even $s$ and $0<t\leq1$ for odd $s$.
Parameters
$s$
—   order of the Clausen function ($s\geq 2$)
$t$
—   angle divided by $\pi$ ($0<t<1$ when $s$ is even, and $0<t\leq 1$ when $s$ is odd)
Formulas
(1)
$\mathrm{Li}_s(z)=\sum_{k=1}^{\infty}z^k/k^s$ is the polylogarithm [3].
(2)
For $m\geq 1$, $\mathrm{Cl}_{2m}(\theta)=\sum_{k=1}^{\infty}\sin(k\theta)/k^{2m}$ and $\mathrm{Cl}_{2m+1}(\theta)=\sum_{k=1}^{\infty}\cos(k\theta)/k^{2m+1}$.
(3)
$\mathrm{Cl}_s(\theta+2\pi)=\mathrm{Cl}_s(\theta)$. Under reflection, $\mathrm{Cl}_s(-\theta)=-\mathrm{Cl}_s(\theta)$ for even $s$, and $\mathrm{Cl}_s(-\theta)=\mathrm{Cl}_s(\theta)$ for odd $s$. These identities reduce every rational multiple of $\pi$ to a value with $0\leq t\leq 1$, with a sign change only in even order.
(4)
If $\alpha=p/q$ is in lowest terms, with $\alpha=t/2$ for the rows in this table, then $\mathrm{Li}_s(e^{2\pi i\alpha})= q^{-s}\sum_{r=1}^{q}e^{2\pi i p r/q}\zeta(s,r/q)$, where $\zeta(s,x)$ is the Hurwitz zeta function.
(5)
For $m\geq 1$, $\mathrm{Cl}_{2m}(\pi/2)=L(2m,\chi_4(3,\cdot))$ and $\mathrm{Cl}_{2m}(2\pi/3)=(\sqrt3/2)L(2m,\chi_3(2,\cdot))$. In the rows listed here, $G$ is Catalan's constant with $\mathrm{Cl}_2(\pi/2)=G$; $\mathrm{Cl}_2(2\pi/3)=2\mathrm{Cl}_2(\pi/3)/3$; $\mathrm{Cl}_3(\pi/3)=\zeta(3)/3$; $\mathrm{Cl}_3(\pi/2)= -3\zeta(3)/32$; $\mathrm{Cl}_3(2\pi/3)=-4\zeta(3)/9$; and $\mathrm{Cl}_3(\pi)=-3\zeta(3)/4$.
(6)
On $0<\theta<2\pi$, $\frac{d}{d\theta}\mathrm{Cl}_2(\theta)=-\log|2\sin(\theta/2)|$; hence $\mathrm{Cl}_2$ has its maximum on $[0,\pi]$ at $\theta=\pi/3$.
Comments
(7)
The index is $t=\theta/\pi$, the same rational parameter used by $\cos(\pi t)$. If the unit circle is parameterised by $u=\theta/(2\pi)$, then $u=t/2$.
(8)
For even $s$, $t=0$ and $t=1$ give $0$ and are omitted. For odd $s$, $t=0$ gives $\zeta(s)$, which belongs to the Riemann zeta table, and $t=1$ gives the alternating value $-(1-2^{1-s})\zeta(s)$.
(9)
The arguments lie on the unit circle and $s>1$, so the series in (1) converges at every listed row. No analytic continuation or branch choice is used for these values.
(10)
The entry $\mathrm{Cl}_2(\pi/3)$ is Gieseking's constant [4]; twice it is the volume of the figure-eight knot complement, the entry $4_1$ in the hyperbolic knot volume table.
(11)
The Lobachevsky function is $\Lambda(\theta)=\frac12\mathrm{Cl}_2(2\theta)$. To read $\Lambda(\pi t)$ from this table, reduce $2t$ modulo $2$ to $u\in[-1,1]$; the value is $\frac12\mathrm{Cl}_2(\pi u)$, using the sign symmetry in (3) when $u<0$ and the omitted zero endpoint when $u=0$ or $|u|=1$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

s = 2
t = QQ(1) / QQ(3)
CBF = ComplexBallField(numberdb.bits(100, losing=96))
z = (CBF.gen(0) * CBF.pi() * CBF(t)).exp()
z.polylog(s).imag()       # Cl_2(pi/3), Gieseking's constant
Links
Similar tables
Hyperbolic volumes of the prime knots with at most ten crossings —   the figure-eight knot complement volume is $2\,\mathrm{Cl}_2(\pi/3)$, and hyperbolic volumes are built from Bloch-Wigner dilogarithms
Values of Dirichlet L-functions at positive integers —   holds the $s=2$ cases of the Dirichlet $L$-value identities at $t=1/2$ and $t=2/3$ in (5)
Values of the Riemann zeta function at rational numbers —   odd-order endpoint and symmetry values include rational multiples of $\zeta(s)$
Values of the Hurwitz zeta function at pairs of rational numbers —   every rational-angle entry is a finite linear combination of Hurwitz zeta values by (4)
$\cos(\pi x)$ for rational $x$ —   uses the same normalisation of the rational angle
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

Each value was computed as a Sage complex ball with arb at numberdb.bits(digits, losing=96) bits by evaluating $\mathrm{Li}_s(e^{i\pi t})$ and taking the real or imaginary part specified in the definition.

more

The 538 entries were checked against the Hurwitz-zeta decomposition (4); the special values in (5) were checked against independent zeta, Hurwitz-zeta and Dirichlet $L$-value computations, and the decimal prefixes for Catalan's constant, Gieseking's constant and $\mathrm{Cl}_2(2\pi/3)$ agreed with [5], [6] and [7].

Table is complete: no (it holds every entry with $s\in\{2,3,4\}$ and $t=a/b$ in lowest terms with $b\leq 24$ that satisfies the stated parameter constraints)


API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.