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Generalized log-sine integrals

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Numbers
$n$
$k$
$t$ 
$\operatorname{Ls}_n^{(k)}(\pi t)$
2
0
1/3:
1.014941606409653625021202554274520285941689307530299792017489106776597476258244022136470354228256695
2
0
1/2:
0.9159655941772190150546035149323841107741493742816721342664981196217630197762547694793565129261151062
2
0
2/3:
0.6766277376064357500141350361830135239611262050201998613449927378510649841721626814243135694855044633
2
0
1:
0
comment: The integral is exactly zero, as follows from the Gamma-ratio generating function (3).
3
0
1/3:
-2.009666081130543900262353754349164503847935370011071794990849691913377448354258724657100992853890772
3
0
1/2:
-2.033576506072054600912068969700518249992376075613046185506487429858439689686915123555411633065963200
3
0
2/3:
-2.144767212569494391482642112215351297471143105786770476300912096629965921054304862752769297431055671
3
0
1:
-2.583856390024985014623026255591782933518774047157092307845378175317199576455475503130558419383573849
3
1
1/3:
0.2614730960666675542455925917636517700815222958719889207329705070721036788822066496575213109565751692
3
1
1/2:
0.1240456529732743262328264691076254726582234427789188859208008874222951821171155288604920458553032269
3
1
2/3:
-0.3191785958886623084501755229326031903162959901376252622406346324271052091494539616713359596186415845
3
1
1:
-2.103599580529289999449541782645037483838726011595873043136475221848216860126047907826297778816336702
4
0
1/3:
6.009497549818888916204788706203270740596963297439568418836063926751510042002802252687623862369056636
4
0
1/2:
6.003109556529006567309305614032688559530453350601135986744455992164043366464996638788851968894075067
4
0
2/3:
5.950668244139784930757810129013984021591629014624001989128732743893312872574104764576824488613740546
4
0
5/6:
5.830507044549023790975395079462744709672138287562057929136693012013993205987551469995370313453063807
4
0
1:
5.664559704244618390805213689878814232251845559194979946374429864326831982539750497678517633998938060
4
1
1/3:
-0.2555485412929076285523897616833313103773717525363660754251471619798618121552565332148258862640124805
4
1
1/2:
-0.2897229585458992193552919184198654929331169927987921286865932604481965798233930103571801132664491690
4
1
2/3:
-0.4976755516066471960582989542950996307350841078148923599175672830361524888045910118109952705249894261
4
1
1:
-1.663765789490125617407691672739629734745328023025191732988259293730109962517299179348064936014043629
4
2
1/3:
0.1205082767606867833928932246574020975977585822556171450353006903117061488737034354503429995897790272
4
2
1/2:
-0.07186957011302177531232354980107443816929056885500752172026827020463293688673640822504279146434280504
4
2
2/3:
-0.8980016106192205781737467631273740823806911470941474068825305514789854938085525461640101604719854304
4
2
1:
-5.664559704244618390805213689878814232251845559194979946374429864326831982539750497678517633998938060
5
0
1/3:
-24.01253312551691461501571396363162679502884841064631502190162078233929982176368144472895858649190016
5
0
1/2:
-24.01433772015983592359467991814460623522190819473543320027555234179026210338733065095551060837656047
5
0
2/3:
-24.03941440856863860664996830970120731117341091554595983531520807658992002307112550368797930314003041
5
0
1:
-24.22655837883478171663368704510531884635713927472260341881815179182693687724444360512452712081915525
5
1
1/3:
0.3790721060126394312318378614313241118915017609261971487369985346568001050321319301203733998861251025
5
1
1/2:
0.3697803051416735629404268726826347568797826222736508789089759820176665410332402658863646102791207776
5
1
2/3:
0.2708016246010793737741869955751992363356462707701333252917782797097793039367781903804130267510772338
5
1
1:
-0.4940720817770173887139563291186403852729295693490984083795487050567957863435303900524985259312648090
5
2
1/3:
-0.07770104444415548913114440353441612682662347193232264628141215682343177216254726049765436222982745825
5
2
1/2:
-0.1268132428355886971002322996610899251838894676273605604707350922797228161276385648999641494175549643
5
2
2/3:
-0.5181087868296801173472656387316967550218796682431532140673894724824649305920679150681759179623426341
5
2
1:
-3.653311323439480953827188154519484128703272043550158769240301907865497825411620537118913458842330063
5
3
1/3:
0.07051813425519439295624239757810105348369969957034380597181438031778620009217081518081895595825909794
5
3
1/2:
-0.2008337702984204851947613904962539860006013666378903937704017234061802143811521080906440213673827373
5
3
2/3:
-1.750368862533376848561949767900774798490094755576787737252409113085453215801501470006964273744485497
5
3
1:
-14.63932357547182220205852967998617301944547911070207537819361085960693622971363454597168791531694466
6
0
1/3:
120.0207613710553001755048886391927614834489250443014689821689519463048640999550204538254628532982064
6
0
1/2:
120.0202330316038894028950652142933511034781044764708260237197403927108986116927419250764821891052205
6
0
2/3:
120.0080913739878069658769876157297868208042703811775593172577862414440327664273866815163617525457012
6
0
1:
119.8852400579281896736967018589288678430302320347399435542106179036819397927446559145343043346344132
6
1
1/3:
-0.7540147822174039296820161899690229594969039001374609922029414784873074226328194681764351235917607915
6
1
1/2:
-0.7566710771286357978500320815104909053715419934107836041316235529201899595186378906316631440245595737
6
1
2/3:
-0.8043989575184219520625703710245259959375977147553456059926488976694959080649791987426451773257099315
6
1
1:
-1.307852477704983146246257087275725495282018028724402009848580628952880840080594524552620682566120447
6
2
1/3:
0.07619665511420732272061296069148858119713571071860673409069863434290962147408359619154972111407688529
6
2
1/2:
0.06262777798157532281549657011246408977326522697239062505455049748987670071368403613309151687973447184
6
2
2/3:
-0.1252437272995168661145371834606490291899812037541152797104457044827260765750253068074609030987507342
6
2
1:
-2.194373200153759126319553651404804379953391292493533624364169235540091200267709075555280596677939971
6
3
1/3:
-0.03382410887400887449993926871323978365460004851435177673609632811648533948641290840236906618925534055
6
3
1/2:
-0.1047626004217939164005921716698559361888823476578201166855737869632192032576802397789496366956374609
6
3
2/3:
-0.8453601172885053949598645532591061984715702067585794270813310058435581743884475807186184474032062847
6
3
1:
-9.373900391104114072361320340437674704454420889131853302314783885608890599304879775588552211605794039
6
4
1/3:
0.04703770024727430335114262596654791991028976501662147511953135474733187426940554704485042651822750153
6
4
1/2:
-0.3384091533325383477130545132811133142335879600894983151474027883802977328247964709322168003286070545
6
4
2/3:
-3.262399693841844658363750258605295039069705680359319613908668274659191489870327587874858774885316395
6
4
1:
-38.51782287260248729178151100220920446897160934288984955902233762169700987099957655262341576363987909
7
0
1/3:
-720.0413727026361868324054384156808323219739773138289507099468607266035542921737876075764787853564314
7
0
1/2:
-720.0415312916800711885482290972010962090373348863725255069339989232792304982856593532929253266621969
7
0
2/3:
-720.0474758896076043510750075799839162607231822119882562691964900019015994841661899140169765548102832
7
0
1:
-720.1283922997705287210497022333626753416278425220058850340806453850483465556345793255085286948099259
7
1
1/3:
1.879972022580670535258612664572810413975793736967671305761692065020827809418977449908866770867289961
7
1
1/2:
1.879187592621966465582288679825880586617990129980078468049720786586733642182079407763495255732362425
7
1
2/3:
1.855903006517072662998722349004947720697933981450714266801228302400337972307954957834470870488181047
7
1
1:
1.523447671403400298378699569740606312559830526973943949368762839233428877456033225608758370309698622
7
2
1/3:
-0.1004340877383886903164338969003485684999179715570982170782564406997509293293800476674528256215852297
7
2
1/2:
-0.1043557008655757028186411228030013084419993238980028267464628049305127464335481245386407581502891319
7
2
2/3:
-0.1956661321868070500768349783384673363593459790449978617688857619461852328244282919085216929961041112
7
2
1:
-1.565482247928882324406589601396357911786888063795248675342131515483188172551904763127463790457482115
7
3
1/3:
0.02468954353505194284037531930975628980960498439533097423969969070668278027524139529702597767664861429
7
3
1/2:
0.004802146180161942428971357850420761609124463618079396024448891215199141755531230799763350662757474894
7
3
2/3:
-0.3537164933407402710333208305971402919094418498595452321810650735099743281915049171302065764881911419
7
3
1:
-6.013889600278666394915465821574175575696205118637680380832234400173675596216777207587827653917003636
7
4
1/3:
-0.01793363725043227140391102135060392828273237706866597411457156371681330666544802455087545435948735286
7
4
1/2:
-0.1208812652428763664528001202932773905484988875701863590144781920379777072156119091099406303196278089
7
4
2/3:
-1.530423829083985867018401393277641930380409529148311110094358006646091253253741138858106888674316005
7
4
1:
-24.98553106998370879838197987890497601738456375392339905484871221996530807065256289902270148849785866
7
5
1/3:
0.03408442659899093640708704353464757932691471688445396758364005663625665151367245168667219018774428929
7
5
1/2:
-0.5169927828738529212880532421733921953094204292151641028072444915084824783961310219674128304355584285
7
5
2/3:
-6.066537903112002763329570167370195433925457873655834612672086275054466721573656377420121058696653105
7
5
1:
-103.5391814439576807539864217667863656975292160872676170645318451231397273810238180846300438768448121
8
0
1/3:
5040.096328747852098313337769156555176136756396583640217967529495119119880320134280598637333950682671
8
0
1/2:
5040.096280259890316263576585369079700931678816964818854503648142319125216063710247020435793142174460
8
0
2/3:
5040.093340869722495509320616830458706321595826119462773435567000693883932074960549152907670831531125
8
0
1:
5040.039879115045164345621438335393159305375961677482002002138539161341199057514062158954245303223357
8
1
1/3:
-5.632412527098793612288767679683601931820824088830107937497242803619943297875902861113870547716706369
8
1
1/2:
-5.632649535119816153417136714937779010282859804453853400995763396812842322455224372125070999046970649
8
1
2/3:
-5.644127505793152720831952934610103417950504670592716794578028689064240042252957552754484791719492193
8
1
1:
-5.864322642422686042718448904252906758320856770761079213212459674247443721380313097594785457910197913
8
2
1/3:
0.1662573464053301594547672726157941365944735345743420575610059820456623935830626346924053279800087415
8
2
1/2:
0.1650899771478242574174996901847768678312839306524971857643920119882596334691167028610093507114945255
8
2
2/3:
0.1202257670164632683730462446154690831828939609296751108534280828396312285539664807668075887262359988
8
2
1:
-0.7892381581374771681249210745131531164066904669434526693776669053042205501315761408493198196301187252
8
3
1/3:
-0.02425542696642448315553800102777250846690540187480545165324553926124990550269040292173656399406681164
8
3
1/2:
-0.03006120056454279916748249163932264365870380134730352780018379829620813759488868397924023639400183978
8
3
2/3:
-0.2056072390645849567907803536949814300710424759371441238014024032076225012473347566660413448308466429
8
3
1:
-3.972465316366203689588361491076485840218743670167514303078898882918065560862222495676382997508241307
8
4
1/3:
0.01040438632126655649620014591552329355696982140356053592327333306456462013958515124564009249329702833
8
4
1/2:
-0.01884323449550176650057435696638936566585628708175231686223898935118825477189134752765489266834964195
8
4
2/3:
-0.7064959334982537365962702691083895063348045025195041080406475129494685976324343092164200498154066858
8
4
1:
-16.35193079207812008886606891024699058081773715133154396652174513536654588661098933292611217768832426
8
5
1/3:
-0.01077348884213641382813643506162212653585566267341337903777440992416186782756367019361102582794716232
8
5
1/2:
-0.1608244922339263022802117931382050667301923363556242120262846188683024026889012474008806632694579022
8
5
2/3:
-2.857740531032905895356625131090672931037555985894396449418157661027075462518003354414238562730658542
8
5
1:
-68.02133196722819191223658349413885157774748314116712299097677265257627979659834144162834033102527924
8
6
1/3:
0.02615021752359178781753328422040709778443633083072781086171150094207132373736358438371053268566804089
8
6
1/2:
-0.7664392590806598832789245713489972933254637524926742343153729558326618127684993205735615164816022712
8
6
2/3:
-11.35704892896107206570706560818257483559461544821348838546295287799987772613931154116348199119125127
8
6
1:
-283.5154133520471793177490688813024956832038008159223515240188475069025386599305205490835226322103171
9
0
1/3:
-40320.25654736190482935106713010874748559231288198813889477292040745376521959858940501751265095816075
9
0
1/2:
-40320.25656239873215988457068436603695707135940532699606721388496527634348591849308478121810936284332
9
0
2/3:
-40320.25802854715200639604050121738558955884411639772354624205801484435052198454381567994643392114880
9
0
1:
-40320.29345280260011746949122188233777673071504944491982198497910397580014876382870268018693682706103
9
1
1/3:
19.70041963598002447406853049530334913224677513511679595868688494007978402018790588880918957294899061
9
1
1/2:
19.70034679633274302421134041590704638220026456707270637419307139404655282993266336763270164194163328
9
1
2/3:
19.69463680953546128103757572440661546653852393132264463656866128282016233328306057645634326865992892
9
1
1:
19.54839089869786756616891110022002200942106092862117030044846787934112621242687966014596872210180465
9
2
1/3:
-0.3311790164714339363884617109412293075441250069546306124521909279728546490207981238497676937715549545
9
2
1/2:
-0.3315338867254081958695153206456129562305697953741716378175470692545190098809723308369190990845717089
9
2
2/3:
-0.3537900828698566382297286403842495754919279906728260180492612025536741589984036055828060713905453395
9
2
1:
-0.9592131184452048137628184224207455099807404153232288629539450348560206638751487338883261046775545918
9
3
1/3:
0.02996834959681887253503937728471830195012382490823112480141747390986675924844298569831468829640218539
9
3
1/2:
0.02822729432370285866204789088979770550533815801556925043452000387532494392336578773368810176969943711
9
3
2/3:
-0.05859875281017237024755963352278882460899147515575787702895063253643431340518936609119238220715066649
9
3
1:
-2.571793310241111051963259456280152990931614057845347036073429612999950890591598044453889747040187392
9
4
1/3:
-0.008129317888029080052750127686227744782952361548366037040308190310385071057939707448199788008363694744
9
4
1/2:
-0.01674684951254045914095661747053680099360642766299351233874379363088148757341582390830131983256643598
9
4
2/3:
-0.3557929692809870147804606810125273154909839993467902988243849127372238155256454666917477665456326212
9
4
1:
-10.81698107801949748465035840379188275341626306034565699871226741937770713672449286226157712562208630
9
5
1/3:
0.005179771315140934997362297613982721725268386794247038164804014022448438838755373927314396139360511902
9
5
1/2:
-0.03796989282970802035924126198579606721940758490422382987224962889029066379259116887602498420063290786
9
5
2/3:
-1.363243359202630110753739622932485512969982399106030746353856596812149651435589063592779348134310606
9
5
1:
-45.02643820538767220527164984878736109989065235617679879915882737487538738234188847025228954626066104
9
6
1/3:
-0.007055020922368515400381662863737438613650596388666659298800192089948556008038645336098714601879973740
9
6
1/2:
-0.2266414767554487454002622073758360967103368735899873848968983419374992680696100516502525118823037314
9
6
2/3:
-5.412498180031304567819803493124155869939922800387239379801463587452096962501937386275226042736394558
9
6
1:
-188.1457985657277956237680708614163206332106452020060995144282376300640194112276877937829043158351728
9
7
1/3:
0.02091914821585940932100374523876875229370991973027548096621398966823418028760699620358575799513521135
9
7
1/2:
-1.125268490651897896797864120529042924484101912959446757073636741725318760594400792142281293976496036
9
7
2/3:
-21.44117365105161618986234236614816803611956601839816716052254472027381249284874462909888596413169915
9
7
1:
-788.2234789838390207797712568753410359034563421643310867092875057807368079382189057797252767862886879
10
0
1/3:
362880.7690528823416935411885790625247402587103101529380841965036137842827985299497042891149084927957
10
0
1/2:
362880.7690481663390373769630973474576779269283512493584694694580425250369232710295835772439026508685
10
0
2/3:
362880.7683112302960233941359084748089912749922732014995884094532485948726347655506623281211659963427
10
0
1:
362880.7447761794889086465290681255960652856414528049530931975261119730550382171390121368761187490724
10
1
1/3:
-78.77577034220259352448730028603163072386286158391294491347992123090853980160613910501798431374854292
10
1
1/2:
-78.77579302547365301951843305561642126574357468076700712229204511199619103933480039390491206039833025
10
1
2/3:
-78.77865654618864964080154552721468073181074628720813223500270928474166900048355642389613506602330443
10
1
1:
-78.87603613062063277636892994237039202688408479681807430189854293629874979475564751577873072104616189
10
2
1/3:
0.7709041239218690223508050772066021904316491555025151018071501652887458284241674607430898792207343171
10
2
1/2:
0.7707945329382195562717099855379149106512469146107481226329257382297514061576596503408491703393000623
10
2
2/3:
0.7596600198934161363965587813765588412412323896284875091934127662709376499271458913112582668284204634
10
2
1:
0.3556516425625313020066309110144933379315894098572247720793727435829154655817169160918821630949233585
10
3
1/3:
-0.04460938924001834501222449109674375425511629314888611970583915297849335962758726444766841709369356313
10
3
1/2:
-0.04514167550204723088722170072549453587229449521554957472038944791365376108051367770238450258980986876
10
3
2/3:
-0.08846911036922098424944038172192480442815668681987815652884175882918661080999781060390471229199892526
10
3
1:
-1.769128059354675861779638146165822612899621211793622955928514566625646852769948863444383234667514402
10
4
1/3:
0.007995406199406534894063918794720337043676668457398751556908003342784929523667354827870870835774631226
10
4
1/2:
0.005393399037810481148290396255103342233310646335677957325006757822644871944620707090630843907697866686
10
4
2/3:
-0.1633386843459847008054300687520339994554924756595819141302573811653575396608784955541105878960700024
10
4
1:
-7.173514987565935920364691242034546007605602982921106317590521642855161858357522093609639355171268044
10
5
1/3:
-0.003354735766448197452590061800932004723453238630608484534733873658388833520719280154108958459761078533
10
5
1/2:
-0.01617683172244329938458488029188313419028434511993174018686723163055932555447714885798841789568900180
10
5
2/3:
-0.6738333118041443216425034131503924245927067216434217959230765692376269041510166751946706654382784090
10
5
1:
-29.99141512716570989812082369464140306167439367542491925795138146874335031046688149137612439016900121
10
6
1/3:
0.002893639127337473296961979745378211392879004592908875478219692434169937297839223475198144506183095231
10
6
1/2:
-0.06095350555946685423030380425689130657929331096364792908406909282311114564973868263583723644205861775
10
6
2/3:
-2.626576856713939905273346707596432223079593323406314326315507084241406856982497006686118377209142636
10
6
1:
-125.5581587099660061678348530707770812928521388513587466565833483496417510938876821247511745517822635
10
7
1/3:
-0.004921013182192483929258484933785902224943089826086135139140199851908882348668256379592622105813468587
10
7
1/2:
-0.3274599279462318282658852925499078814855567697291122447643858767218593537538193162566801548022678060
10
7
2/3:
-10.34604954682001361650951852081194852648949938266006566362275529098106109118912175054075026767161095
10
7
1:
-527.1405830522868247657318134332957785824154715571936495589262601989352203139328788853142897482000998
10
8
1/3:
0.01727693357780434120587840991675368846993175535593182067435870985878182342965156379584619479944537931
10
8
1/2:
-1.648559453788950481420334511920730189606579967618735988162237069555007189269914444456137125940861874
10
8
2/3:
-40.81099590818971962351067622239810634984117398346055357493770886520142298207329979898144056098507632
10
8
1:
-2218.879072823033673891002146433375509509503007917280027842384616358429754010103276947666391220240395
Definition
The generalized log-sine integral is $\operatorname{Ls}_n^{(k)}(\pi t)=-\int_0^{\pi t}v^k\log^{n-1-k}|2\sin(v/2)|\,dv$, with natural logarithms and radian angles [1], equation (1). The endpoint integral is an ordinary convergent improper integral.
Parameters
$n$
—   total order ($n\geq1$)
$k$
—   angular moment order ($0\leq k<n$)
$t$
—   angle divided by $\pi$ ($0<t\leq1$)
Formulas
(1)
$\operatorname{Ls}_n^{(n-1)}(\theta)=-\theta^n/n$ for $n\geq1$. This zero-logarithm boundary is not enumerated.
(2)
$\operatorname{Ls}_2(\theta)=\operatorname{Cl}_2(\theta)$, where $\operatorname{Cl}_2(\theta)=\sum_{j=1}^{\infty}\sin(j\theta)/j^2$ is the Clausen function. In particular, $\operatorname{Ls}_2(\pi)=0$.
(3)
$-\dfrac1\pi\sum_{m=0}^{\infty}\operatorname{Ls}_{m+1}(\pi)\dfrac{s^m}{m!}=\dfrac{\Gamma(1+s)}{\Gamma(1+s/2)^2}$ for $|s|<1$ [1], equation (8). Here $\Gamma$ is the Gamma function.
(4)
$\operatorname{Ls}_3(\pi)=-\pi^3/12$ and $\operatorname{Ls}_4(\pi)=3\pi\zeta(3)/2$ [1], Example 1, with $\zeta$ the Riemann zeta function.
(5)
$m_r(\Phi_1)=-\operatorname{Ls}_{r+1}(\pi)/\pi$ for $r\geq1$, where $\Phi_1(x)=x-1$ and $m_r(P)=\int_0^1\log^r|P(e^{2\pi i v})|\,dv$. The higher Mahler-measure table contains $r=2,3$.
(6)
$M_{3,r}=[\operatorname{Ls}_{r+1}(\pi/3)-\operatorname{Ls}_{r+1}(\pi)]/\pi$ for $r\geq1$ [1], equation (3). Here $M_{3,r}=\mathbb{E}\prod_{j=1}^r\log|1+x+y_j|$, with independent Haar-uniform unit-circle variables $x,y_1,\ldots,y_r$ and the same $x$ in every factor, as in the multiple Mahler-measure table.
Comments
(7)
The exponent $k$ is the angular moment order, and $n-1-k$ is the logarithmic power. The notation $\operatorname{Ls}_n=\operatorname{Ls}_n^{(0)}$ denotes the basic integrals. The absolute value encloses $2\sin(v/2)$, not its logarithm; the leading minus sign and all powers of $\pi$ and factorials are retained in the stored value.
(8)
The rational parameter is $t=\theta/\pi$. The logarithm changes sign at $\theta=\pi/3$, and $\theta=\pi$ is the half-period endpoint. The angles $\pi/2$ and $2\pi/3$ are also distinguished in the special-value and Feynman-integral literature [1] [2]. The additional value $\operatorname{Ls}_4(5\pi/6)$ occurs in the Apéry-like series identity in [3], equation (3.5).
(9)
Only the boundary $n=2,k=0$ is the ordinary Clausen function $\operatorname{Cl}_2$. Higher logarithmic powers and angular moments are not the higher-order ordinary Clausen functions.
(10)
Generalized log-sine integrals enter expansions of massive Feynman diagrams and inverse binomial sums [2]. The angle normalization in that correspondence is $u=4\sin^2(\theta/2)$; the hypergeometric argument used in that reference is $z=u/4$.
Programs
(P1)
Python (mpmath)
import mpmath as mp

def log_sine(n, k, t, dps=130):
    with mp.workdps(dps):
        theta = mp.pi * mp.mpf(t)
        if k == n-1:
            return -theta**n/n
        return -mp.quad(lambda v:
            v**k * mp.log(2*mp.sin(v/2))**(n-1-k),
            [0, theta/2, theta])

log_sine(4, 0, '5/6')
References
[1]
Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, Proceedings of ISSAC 2011, 43–50. Definition (1), equation (8) and section 5. (arXiv) (doi)
[2]
A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums. Introduction, equations (1.2)–(1.4), and section 4. (arXiv)
[3]
Steven Charlton, Herbert Gangl, Li Lai, Ce Xu and Jianqiang Zhao, On two conjectures of Sun concerning Apéry-like series. Definition 2.1 and equation (3.5). (arXiv)
Links
Similar tables
Values_of_the_Clausen_functions_at_rational_multiples_of_pi —   The $n=2,k=0$ boundary equals $\operatorname{Cl}_2(\pi t)$; higher logarithmic powers are different functions.
Higher_Mahler_measure_of_cyclotomic_polynomials —   The linear cyclotomic moments satisfy $m_r(\Phi_1)=-\operatorname{Ls}_{r+1}(\pi)/\pi$.
Multiple_Mahler_measures_of_1_x_y_j_and_1_x_y_j_z_j —   The three-term multiple moments are differences of basic log-sine values at $\pi/3$ and $\pi$, divided by $\pi$.
Values_of_the_Riemann_zeta_function_at_rational_numbers —   Zeta values occur in the Gamma-ratio coefficients and the low-order endpoint evaluations.
Inverse binomial sums —   For $0<u\leq4$, the substitution $u=4\sin^2(\theta/2)$ expresses $F_s(u)$ as a finite triangular combination of $\operatorname{Ls}_{i+2}^{(1)}(\theta)$, including the elementary boundary $\operatorname{Ls}_2^{(1)}(\theta)=-\theta^2/2$. Its four positive slices correspond to $t=1/3,1/2,2/3,1$; the alternating slice $u=-1$ has no real-angle correspondence.
Log-sine-cosine integrals —   The mixed moments have pure-power boundaries $\operatorname{Lsc}_{m,1}(\theta)=\operatorname{Ls}_m(\theta)$ and $\operatorname{Lsc}_{1,n}(\theta)=\operatorname{Ls}_n(\pi)-\operatorname{Ls}_n(\pi-\theta)$. The exceptional-angle harmonic-binomial identity also combines mixed moments with $\operatorname{Ls}_4(5\pi/6)$.
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it holds every $2\leq n\leq10$ and $0\leq k\leq n-2$ at $t=1/3,1/2,2/3,1$, together with $(n,k,t)=(4,0,5/6)$. The arguments $t=1/3$ and $t=1$ select the logarithm's zero and the half-period endpoint; $t=1/2$ and $t=2/3$ are special-value application families, and the single $t=5/6$ case occurs in an Apéry-like series identity. The noncanonical order cutoff includes every angular moment with a nonzero logarithmic power through order ten, both parities, and the basic integrals before extending any one slice more deeply. The attached argument-selection.md records the sources and coverage rationale)
How they were obtained:

The Euler sine product [4] gives a convergent series for $\log(2\sin(v/2))-\log v$. Its integer powers are multiplied as polynomials and integrated term by term, treating the singular logarithmic endpoint analytically. Arb encloses all coefficients, arguments and arithmetic; a positive-coefficient majorant supplies a proved bound for every omitted term, which is added to the output radius. The attached certification.md derives the bound. Computation uses 525 working bits and degree 532; the minimum measured relative accuracy was 507 bits (more than 152 decimal digits), recorded in numerical-evidence.json. Every nonzero enclosure must retain at least 399 accurate bits before a 100-significant-digit decimal is written. The exact value $\operatorname{Ls}_2(\pi)=0$ follows from the Gamma-ratio generating function.

more

Every entry is recomputed at 140 requested digits and independently checked by mpmath tanh-sinh quadrature at 130 and 170 working digits. Nonzero elementary integrals and a deliberate wrong-answer control are checked first. All nine basic endpoint values are compared with Gamma-ratio coefficients, every one-logarithm angular moment with a finite polylogarithm expression, and the Clausen and Mahler-measure relations with the existing companion values. The script checks read-back decimal intervals and repeats the applicable identities on the stored entries. Quadrature is a validation check; the series remainder and ball arithmetic certify the precision.



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