NumberDB$^\beta$ Advanced search Tags Tables Help Login
Values of the derivative $\zeta'(s)$ of the Riemann zeta function at rational numbers
edit · history · discussion · files · short url · L-function special values
Numbers
$s$ 
$\zeta'(s)$
0:
-0.9189385332046727417803297364056176398613974736377834128171515404827656959272603976947432986359541976
comment: $-\frac12\log(2\pi)$.
-1:
-0.1654211437004509292139196602427806427640363803352017836665223063573596996665771727595251003325087555
comment: $\frac1{12}-\log A$, where $A$ is the Glaisher-Kinkelin constant [3].
2:
-0.9375482543158437537025740945678649778978602886148299258854334803604438113127075227936894151411515175
comment: $\frac{\pi^2}{6}(\gamma+\log(2\pi)-12\log A)$ [3].
-2:
-0.03044845705839327078025153047115477664700048354497393625297188985903781794493689867779458488087449597
comment: $-\zeta(3)/(4\pi^2)$.
3:
-0.1981262428856368533306818215032857968755427934638350033468899631927256694226511041515708421775957421
-3:
0.005378576357774301144416974210413842895664439742295507059447023223324501997924069586095103870825683267
4:
-0.06891126589612537984882936558744082715001637487137846382758570601842849852762120133478041038984760229
-4:
0.007983811450268624280696670798789303905237693362298876417704739714028740281878657952543961969286982040
comment: $3\zeta(5)/(4\pi^4)$.
5:
-0.02857378050946295008038981708380121264821523555636624823848001794536450116616395297676555989701173090
-5:
-0.0005729859801986352049909941488338745132539872911995212178207918809977350313508312578653993423857005060
6:
-0.01285216513179572507594540145992360882884875391193195136796944988339867117171289588221630843096867868
-6:
-0.005899759143515937450629877408392025579801534620157195865239392206359703759424905723023863007542258385
7:
-0.006033516960875637793965365478530432397640013548691334714307692521376137030039495342180001675585284016
-7:
-0.0007286426801592406524672333546503606119028877209254183186363861541425975455273099130232464416804493796
8:
-0.002901952553710673130400106656218756390516301366166128158831797751028933952751176138154515847293788321
-8:
0.008316161985602247359524426510534214225674122918829999040210532753056917407881235383483452514524403517
9:
-0.001415982227241809104982959548421754526855060226616749255422629547458870215821469673016811648911005393
-9:
0.003130145319788572754925768290785446702669365865481515964905133205434716301429643494309512677193955374
10:
-0.0006970330081713936936800225787050100786491154573160742715773178387415023556705377809455761373330681018
-10:
-0.01892992633814037422898050229034679523198525809516955581048623110070270515504148055235160734739912371
11:
-0.0003450222223683634969283285180435976709021744972706416521514764461619039553839832743090506204622345612
-11:
-0.01275298447996665611352252548872579815623893704987429279324636666114070232062124740904819354238269507
12:
-0.0001713828458543537668252540843728180429233320609483181123285479605049216495926632278858954557645130043
-12:
0.06327058334146300059518230123430776751141818475323637667956594567062152546067497673747103437135461765
13:
-0.00008532390865593045869859217525223392837925463034261600976910026190237171323790888359709808780850632522
-13:
0.06374987374457688028603868147333505564882735531275849138510088588773706420156668709470926781535826319
14:
-0.00004254149338178082305356123471230827986529761588335445604555490236326423621537672085415187469466928331
-14:
-0.2916577247438735203212240030702506669702630385330908321499093596565151870284637586775093924097273269
15:
-0.00002123108553300034139343867384249695650982773176632319953615798493706996123209927108924197122495944825
-15:
-0.4003193028077255938435803175203203672012612862662329442841069426390303360293172007642614642226439548
16:
-0.00001060244203251431397849056293995757374061949870599031149984324332693163436060808928837934629985215285
-16:
1.773025660899096396247787344189294481355419827646999177163917307737280926906655310456023712750519855
17:
-0.000005296883357529712470385007189405791699306029415273028169159436426095807550833786662574481683115711172
-17:
3.128645332124157875684452639153330548226339077565479742491657706114341129693400534711736286663060260
18:
-0.000002647002978815878859539508872232470270281855523877440612889965464918478261376130301585061592928254452
-18:
-13.74276825021405443522056419051855107309537215770498560474565153488894633788585388234060990032339560
19:
-0.000001323023694778489959165608947957860631249057959246551401609791247212768104160186996102468261715328538
-19:
-29.96552983139235193943186529727420179190822610911556591588187166820576160286767761172687363034256456
20:
-6.613530207367107732987347239762707863211971594369636878202540687057127462521285445777928354141119415e-7
-20:
132.2809975042125145270982115857855186806480099995503145884745019241429157199404293877839464496253429
1/2:
-3.922646139209151727471531446714599513730323971506505209568298485254720803150338284880650523104145691
-1/2:
-0.3608543395999476073474208063639510658848527879186322108143762812758081061266565103095733085083091603
3/2:
-3.932239737431101510706388578406015202692743554892577261544659942554663642773167577863593928580867577
-3/2:
-0.07630925532055088662031899802550575518207158928525969563453814325875965349389545843514686088308548910
5/2:
-0.3873419503262099727119923759310510131994822887468830534204174519487985725767833576462701870860168091
-5/2:
-0.006265736372189758403282758729725833176322370751750072668548840861911795133091714752372914705301634564
7/2:
-0.1131191696081024719011983211701014236517368222257128431491062793203297824358256300698767556696607991
-7/2:
0.009154213629941512461160751534496711060776157027153352355592589021357221464965669518680151213658490914
9/2:
-0.04375753919162248404946701049688500978490789577836491542478001881208180237628250875197041634285145287
-9/2:
0.004097858814523407775962493451265816103077622384663995643618559114253973195440253793152575199142419725
11/2:
-0.01902833981333391068417461747874587776543178135925824416990191472347738733698654227538691505628326064
-11/2:
-0.004339595413423215368019577138401297588435345871794723679533861254176152602014881061636168098048525997
13/2:
-0.008771572542939033388926323573377834161959844263649178955238764256492213260541173759571196615474454303
-13/2:
-0.004602281324001570850842828272395050497599827828078341088557473889667331329953149342136879492947807806
15/2:
-0.004174847373018551039077508392291511936037006159890750879811517704650533974507076261127759736851570300
-15/2:
0.004349787661844826824295806798823139528708211558015238185351907812861265400611446465681640576929225826
17/2:
-0.002024299094422239007153543165807792605393842207553822524657060045923980776106250929664328224284747933
-17/2:
0.008521468121185258350182700593802699927767097827618955336746404016740038987594418125922878231172945477
19/2:
-0.0009926204760338182139931956838492263041186090171474250484883130895589771000258873174701627669858895622
-19/2:
-0.007380504448811925130525772456716747378384968066980150112503906865787343112647792437249201401667216294
21/2:
-0.0004901337872918894082412043951227162001188455088125056134698256448722042440682220349484433193935829984
-21/2:
-0.02375144392688431906708918063992627638323297381533773577588257346692256307898404468015058490428519732
23/2:
-0.0002430838966573070963871707701840755673962968343989713515487623288140247018627248310872742363598180000
-23/2:
0.01883862138459649956537828367203978769685535972103707516208640560178372343976873759161935843764424926
25/2:
-0.0001208986018398176298013341764006922495189444810218733798736015179532169506224167124443550702408519074
-25/2:
0.09324349382377824611200660927351962254379521305060269274056417254050720229672343480182047506244091270
27/2:
-0.00006023903175237361566619166098305787990558668353461104658761403057482377243601044473915021676689074523
-27/2:
-0.06707558118086077548437995812342206989601891418141641093158773214517773644691302461636869479873289298
29/2:
-0.00003005042825299870732940704474543946632348817199424569400018315211937511175201223616408409954510432941
-29/2:
-0.4910939185023732804067881497388982365461526936583612410320253232306838003051691406349656792645443533
31/2:
-0.00001500242745239694023248358001367928246544017322244748959234547651385563384434934639080197525345500397
-31/2:
0.3155935162754465775395816372975287802963493713004044631960805533172657793575829258268574608250257448
33/2:
-0.000007493677289634624203052564733790826857673316207283715014420128669482531379599099442018996171094697226
-33/2:
3.344441971509291358506294507605445229815649266527536613240108217494198056569987473546385701530224682
35/2:
-0.000003744340990529012774756054791191344264807219006672296063340681184841947162112214456300780648828423406
-35/2:
-1.880850435030721874532514401964093424115153531786559068284956102150770363483580095437182094870809902
37/2:
-0.000001871341510880451192678684089222277317749928579411417590121755482184376357610455320987551006087347799
-37/2:
-28.61502612574062394768036949544169237818212693744398121888115598536638236197965270560538999848696557
39/2:
-9.353953082461457757245478692226268055986523938640841515620307401308971475613897462646677755513062137e-7
-39/2:
13.69312757222295547545480238495239853620795799449103110051539961350559653680336732185081209142303696
1/3:
-2.171301352116401570701113995997899337280683009404155270333821276049576450894291928578042030054829274
-1/3:
-0.4795994136588073798923214472235926668263247430724687327281506710825473236470544432759197005189899313
2/3:
-8.924082888363975778968038973764120029257124182404336401173922899485960430468108703154429023767947977
-2/3:
-0.2758916572543389741865722521883085230193042992409571887364085856315320665656916983244765865702217141
4/3:
-8.930514438660476776359822713509827160899070494701231218929760633599384656122524260398865807924454286
-4/3:
-0.09945975069914963419074913642011607932326314942368086913887872334799875490329719067465892756406541972
5/3:
-2.183992702390828667534754527534215201999306375391829037246208243587410432057781784610179276271119640
-5/3:
-0.05769611258748921504155456508352153977440875189777124623895128019996510334846613610733898044013608724
7/3:
-0.5036208538094433264340206519334633659757171070165454047607829407885174645505893378334493944941117918
-7/3:
-0.01261560489503012601007690178706923151760041773580710403606594148791527123041240112734704820763475237
8/3:
-0.3046603874001167406909333094047884247597605085718979893711004179948834931713433345591289743453337311
-8/3:
-0.001276548311982151064843924494119236478267121538986164011037566859588246805437815937298381572645099134
10/3:
-0.1351599561390635933847923378149017496850775796146128701089809637859078738288628054500347128651815171
-10/3:
0.008569069496466135138907550362693702576136446241433160682551386958110740018533453243462265738203987848
11/3:
-0.09534193073860813213547138604065195486138080943845147622697817926355236298073886616581069064644756983
-11/3:
0.009194874985353989763901118383564503535708942350975992583006294394763927518780891875469863300923637184
13/3:
-0.05072453683635531108207443260789897112081041487255947936527463685911893698657508052653366451425823394
-13/3:
0.005571250435827263293401365984341634927166354530161230378724724621286997117587227201878736131220072959
14/3:
-0.03786349091253958637772451384044378553047904293583849343800363826756238507163489473118293341393568589
-14/3:
0.002539033173311552846508767905757580178940932302013839996788411413736353792853737363225550838375961396
16/3:
-0.02175083075190119245106684732039397686043041842497808688254254535262327132253377663522582162139250488
-16/3:
-0.003275458947887296305560421743770935249566893728788452890834013733862867589251744412516682262919070385
17/3:
-0.01667256341249290998961505827069906880436907340803311471638628151562387886252219617421578597214557678
-17/3:
-0.005153239597349809427571786592810072235934296669257165857604822478873392460848095933532414169500752356
19/3:
-0.009952953964604688941122406907265908819843831391546941929772401922145761993630656411222855397646837649
-19/3:
-0.005356737028654339548145897778385499318618080002200893909834656482410125749641129724897263095034891395
20/3:
-0.007737082736410671913098685283818514813489783344818108293124378178025697091152266503287451164921344808
-20/3:
-0.003552293313394800870509874686359070706252457232243725169814559098265222693429605325309952514729636588
22/3:
-0.004717422446544018977970704023116932495727932304757940014725605585813801317644944098011092903755023403
-22/3:
0.002651104178185667638567047273515675363606124532920263529914405849977678447143177837656915903749252467
23/3:
-0.003696537912652176246009081922743581108377056870804638331763419505145948807589410009616100777039102936
-23/3:
0.005928251656808072337711550937168855996659659015810968373492829971990246915397406411954113448467321655
25/3:
-0.002281735575775211866367631756729368661180995748994381904112096915252884049887518439054865154435084327
-25/3:
0.009016759750848830524753264626732594987910066645810156391997077457455194074396326617800368878799354213
26/3:
-0.001796456051963272037793140200156842435855791017862190094322088788686306681471075657081960445980445680
-26/3:
0.007385150912261860426033310731173994970977361728137758586839260278738158015077524012481520708003307690
28/3:
-0.001117163180536984496975135037522495968988922340132069024455649753615436803012179745139377171652646439
-28/3:
-0.003477085080197585986432154405933675997710653946127036341172163297295324208863537290773437803758480862
29/3:
-0.0008821292642059641326523571651799158734633310119653331087255351982590794964014836631713858426642616960
-29/3:
-0.01143233799516276683146732026953956598475561182858378467981483694774435796447711778309847276328908784
31/3:
-0.0005511077589978436062838579574825427696198542477805535176472063801037968676881603859351788490886990071
-31/3:
-0.02347316821645930438119467485685955177147466421836996402126937314237976325517707586168222001530480426
32/3:
-0.0004359582460671609811504127904856119033412724758157172192168110536773538232855058231808194472294642938
-32/3:
-0.02223965657267089411861931731378571557721446371971243734100382509484508623368093751560859679306225776
34/3:
-0.0002731594666498610115610552086411517438038905650365934549289403297750221413497238483401125209569640378
-34/3:
0.006145224087297846953119759738307671513960289298748486705184025291145656654887922385710078474165980712
35/3:
-0.0002163363795348833942398849388023362467980336167197771852868984168481441990655310140020849984759930804
-35/3:
0.03312849023687047806090237582564636698952583913917986126034053340655684792594373259612696722008276246
37/3:
-0.0001358040724923695604025635341741730862605440563862458418940634593269139763051999735699750892320810596
-37/3:
0.08739808750145077210470687198888841228300813229991748483060928762110298361281330835323878481515686657
38/3:
-0.0001076344787133444145388475389843760220834714506725207195559257078303865316067705985264387571526758006
-38/3:
0.09248538806735308559718217644853661188609315124805341969854114227105777360298182229585836964671538753
40/3:
-0.00006764879145958355635866205460347319447457029132484121387581510157286272443812172179252114479753283792
-40/3:
-0.01098875090296362997795336813381057901972351795833121860966317490666014219857001678058269181608534873
41/3:
-0.00005364262747461159915760132273849042011316192593359815499591350401154610464623640053978794519477817758
-41/3:
-0.1345170271455394087510497697336104025516229840803797925144869121972910038521113528444557117858522643
43/3:
-0.00003374125127940377269440992551678190129890540718458154095462520432898649984959755069933245369790430117
-43/3:
-0.4408901191771648339299616685010290791888095643040833550152875813996653489997684154871896717445629232
44/3:
-0.00002676390039531340554378784596965084268461859365178767355460382007659736514053669039112243188758150295
-44/3:
-0.5098632410054096938231267151060579920938571038508283818015734354465060152837388908333875291820603740
46/3:
-0.00001684321788260147662842164117615170976976153828340636461876450672513705666882549688382649472016033521
-46/3:
-0.009109030028375037077121306204042541063506600148956170907559698195616216091587796451192611113407843275
47/3:
-0.00001336300402206074116934792188184984649040932082323972077504736815003915971202544231772529471590854927
-47/3:
0.7287212358467238880698025842429960519346275586911422857468884064172451363556220146456810753622462273
49/3:
-0.000008412547793334058343763399094967722959900778338377410025902887516500397264500044919087309091295568669
-49/3:
2.895678255755942309424624420829455628386863115932404532066075798288006451070669094744474070860077803
50/3:
-0.000006675233373595876972338813008903184006963884928993654495669548370092150297373561145079188236204613758
-50/3:
3.605723862652982628738928351874355765475185453743738133057408021924410933627441997262002652262459431
52/3:
-0.000004203271826097758511805841564325671151253165485712620107168548315713326723219962671345855217407796299
-52/3:
0.5092733366083161219027172306497232137109298361731893562900501073404423871553279258792253478357930157
53/3:
-0.000003335538609939060935014658583422231900420277018102247268572099846759329223127215854709026596500692191
-53/3:
-5.078069816020962278493699997022422017759543749911940209802567758616627312022710745096859215477245110
55/3:
-0.000002100639727556153101654350668617734233590821812133049831489683247576711643781828868265851084540467947
-55/3:
-24.01424335941471485952084929464064771815237786238709736579060792790561708035807528987531850928522311
56/3:
-0.000001667079437506147510424355616020089133502520499740190495538362900671604305502092021286371170207437329
-56/3:
-31.85534899094282104397298049738808244582350955109788489486269229701518947935640971508109397433065772
58/3:
-0.000001049988913898670213789576348083183235139562294509608061286453890935325028171670567970141211827490977
-58/3:
-8.134151947814134123052732817107863991673539892108624850455300580662315559012948556532270576827805484
59/3:
-8.333104608938205457157672796273522960091758546380841133386072123870347818062894570203875102461104961e-7
-59/3:
44.24863955908020346179805318477224366860473335037635366502341134511971468350507363064652452536213911
1/4:
-1.698445150389634033826359313094341485813470858174917876703159318752637434529100637564044192228748069
-1/4:
-0.5575042169665694383724323492264591616909236033987121951169321839432939248841634306006526186710383591
3/4:
-15.92483192869048636323051393774518207702964074897147062761671274395158065853008725667097405614264908
-3/4:
-0.2422315039543977171349444365649688465412949919404645610526523700181635993663278574111251176617769936
5/4:
-15.92966500287924224283348267520329362684696668758653877640120190131138682861102525332662648040618915
-5/4:
-0.1131426057012407688506548991539548365407021403041145807992504004532878067326734629221018676190464941
7/4:
-1.712654652931478990807559047739085189528453979966867806037141410658302826081257958509242913069417707
-7/4:
-0.04977527246137198834076149402899609011405978301341956403570634704475326116282221470501878796594008957
9/4:
-0.5802290143462540128447901125789236066480178633473064814122182968953394321766617409254497868170470237
-9/4:
-0.01637546888274229674876244332011528846998037767601187981172698659749446050112053656357501691504196413
11/4:
-0.2720657596476115053475156699564032497189686675738217695555233327965320212102210319854539023370439795
-11/4:
0.0007713116027178802584262010638582832230167151026462847420256066276779143720700594438974793829798303923
13/4:
-0.1481886370950863970040680613154215440416053367112823179178717281572876926765376105199297461578488877
-13/4:
0.008043265099076913215125703332094524222180908067579742674663431748246589328876573093804202624970019776
15/4:
-0.08773348932193847150603950409831088167914569418906821858565301606569695930924632845965470545569248954
-15/4:
0.009038330803318032557160326694502905869448378874626932018942239391125674945678374560913959206989657375
17/4:
-0.05468333306942743681542501984397564779049432570375410153276420350468873665012833218545456744849347783
-17/4:
0.006254485189464772535304766827462208094246737047780168876048862753028395492017223779417248335689319815
19/4:
-0.03525786217409292295234251887354761291900625524485028974553537643252250456232273704123071031920847403
-19/4:
0.001748395661292463355808229923632395454848083696291190050595167979335159043596504719327044736389456405
21/4:
-0.02326966842444787952201780531873508400444043331794380688668224429427049931864370900795533153678999343
-21/4:
-0.002662501596151104743541670997572244986238109244643564836038111522684503310111457274516978418474312404
23/4:
-0.01561477309397367746066381147770193146136048755872489806582943989896448339297554639351652996815375418
-23/4:
-0.005454953973763473867649694099167374478035165210645705152771031607768975625617192626234011670039631457
25/4:
-0.01060574115554301975420176113478174187115457327242997510990618844708566397397686146503334795950864289
-25/4:
-0.005615350605003363471908121217747372155940062759155165376656021065968332605047370459431008041438179476
27/4:
-0.007268704378259132171348713971358959824761473374836616924605126907012072858172900681483280676315451336
-27/4:
-0.002927552955609380421937882255064351257583710707468642718909239405762306699844725359118775977857816576
29/4:
-0.005015624137241903857154181840766164662726460341463550402745285735448116033306985988051724732891128460
-29/4:
0.001787183462352529075800687325068059235192318881243512524977433113528458249484905812032933331653356307
31/4:
-0.003478954846812265835355253889852352603608000844264466118040909743358358885370168842446655679134982198
-31/4:
0.006640680587189153587056040653419585873535697648459310792043483141939454195223482471394632171725963632
33/4:
-0.002422779064045309561150414582728244280456756061511900530933668229411434217252591026229561243441830758
-33/4:
0.009039478599806722839860702233319625290158505144062630309150909976153389878471851321715567588666278879
35/4:
-0.001692520788361476531842728831153922351521157066250883311916867155774418078296319994281999773678096032
-35/4:
0.006567899527998537693813690803019954924566825473506520585571343216248059277052257759974392704804972543
37/4:
-0.001185267187375326616577362194244280884121741640682685286719173449475760412080039353553680991702780803
-37/4:
-0.001646003472790701119314484528809476777201007525396654299095363638867600083409058624292414998145913798
39/4:
-0.0008316406096150236868208053060545819557692041251836653346133156488299367810092432983043482032739493541
-39/4:
-0.01343715625653142857249293827048978737343271973276444767583505283147484153876157173580978134755050066
41/4:
-0.0005844108622980123881743540782394453535008607533844173570873976634041320721106842948109572933405943651
-41/4:
-0.02276829650918302126408821078271750891714221337195075953082086799996293302748789851339271654851346619
43/4:
-0.0004111765987053139161406181177766416217601083699079335458437648017166244962562989208878338740607724191
-43/4:
-0.02071806457209378615200956285305162331980852382715308752533012819065582618926753929951360316293822639
45/4:
-0.0002895740325971314285674305001785637135997547857897837341545781547660790039895590282903070438352851284
-45/4:
0.0005571039142208363712609505735254003853435448326349728967600472883025550003092740996107087762173535949
47/4:
-0.0002040929656345146328701257377193865099502272407935794741667379715447384610417664605822614819535128476
-47/4:
0.04065193254123815415597852942517094334099126512089945685903909636365468108050895662680650360297322138
49/4:
-0.0001439352508972478704259652561787384381373641128675909740341189976614440821750074834731031121385147976
-49/4:
0.08258904864877210144522628726968941807256448705453208302896346697928794864060652309551358765981962329
51/4:
-0.0001015603349607794203390779880202397239114326366717731010683723596926964468232550073802009806079011766
-51/4:
0.08903658521449233964686311472411674540389849488515321517076364282361249794501643652966578561161466600
53/4:
-0.00007168968756314002976934476450091802760097575963859997545608940726019557424894952687334751838262652604
-53/4:
0.01239841565658113246029957824884809986645200538798825343748840179038057326021103103424019042758853473
55/4:
-0.00005062103988441981324334485931789999318612012689753607490155632819625557176786133342517465420607570672
-55/4:
-0.1717667981464402361814205431877625556237296832064994555985653653178254709242789066861959601679004761
57/4:
-0.00003575362849007066967798230940979747303339711531245643248037013810734795223121529556397980246600015199
-57/4:
-0.4077541179662286584776646853240454571437525851455471540988454571674868042541795152693281083076507577
59/4:
-0.00002525817973170350270068608206199116719086587941668373250770733586461719417446868578405005733284126643
-59/4:
-0.5034047260194403230680674368311754112657111890173906339597918446733337948248980147822184731638118344
61/4:
-0.00001784675467461734488726086823981919809282263177961884106525406037038872363209060240398290482369590916
-61/4:
-0.1380725277342777811127449404891031451014102600292428339816834198474936380723491463922032499347549179
63/4:
-0.00001261181344505006104276411200110436157448060999342301062745494137317677341072417619120996465461174252
-63/4:
0.9661032872891713320129783796453573433043538999905194004117579885409104689083209595954640565133236418
65/4:
-0.000008913441572250210043888025724300633908012304992529150560270436690237160599952156796825159564548338383
-65/4:
2.629527169363759050486671332989937718140223858146744300505808821449087701934995626052476280805495004
67/4:
-0.000006300189495676821427218168648362605698675446424000881315673797448398400768297944206330462796743662814
-67/4:
3.633486545722451272084436633030105939733531358356981767257917216765366457099471671091979982766161939
69/4:
-0.000004453429450528057184465930793642167079767215538276216440481353279906535629178016978862739112562686127
-69/4:
1.416781111549904055923062975242912389287115504178645115200363900343877364663624233322226705693955429
71/4:
-0.000003148199209107945840759624569555706173722284930660111371259248288102158024606874990235664568116919208
-71/4:
-6.978593783187468074522282801985756532994443399421592500548089874782152423618802291567340033651963908
73/4:
-0.000002225622675832846218881314974199283061227225903225548534048960659078001544838866820995910114173085727
-73/4:
-21.46394219456880573197859831200851990379713156137170027066498046310526452620803702639042088125650650
75/4:
-0.000001573470272203816737754304779260687161102388800061123978463895043963912056789579427957968499164643809
-75/4:
-32.65393048068479959603233714013210327162419079285361624161948649079372562647549144684274484566195931
77/4:
-0.000001112448570746074325076381158659033903231771792298067732889184681999838948775888312827416359460270591
-77/4:
-16.07630473288481581863826298169988516369075952856047778387406893199284295691319979738003692565957965
79/4:
-7.865259799314143864317581891556846432148992078583562064362003051852894632072899507700449258689576117e-7
-79/4:
62.96322279445002152866508961787209738190386858983356565505197640360194249979541495421709714466644600
Definition
$\zeta(s)$ is the Riemann zeta function, $\sum_{n\geq1}n^{-s}$ for $\Re(s)>1$ continued meromorphically away from its pole at $s=1$ [1]. The table holds its derivative $\zeta'(s)$ at rational $s$ where it is finite.
Parameters
$s$
—   argument ($s\in\mathbb{Q}$; $s\neq1$)
Formulas
(1)
$\zeta'(0)=-\frac12\log(2\pi)$ [2].
(2)
For $m\geq1$, $\zeta'(-2m)=(-1)^m\frac{(2m)!}{2(2\pi)^{2m}}\zeta(2m+1)$ [2].
(3)
For $k\geq1$, $\zeta'(-k)=\frac{B_{k+1}}{k+1}H_k-\log A_k$, where $H_k=\sum_{j=1}^k 1/j$ and $A_k$ are the Bendersky constants; the $k=1$ member is the Glaisher-Kinkelin constant.
(4)
For $n\geq1$, $\zeta'(2n)=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\,\zeta'(1-2n)-(\psi(2n)-\log(2\pi))B_{2n}\right)$ [2].
Comments
(5)
The derivative $\zeta'(s)$ is finite at the trivial zeros $s=-2,-4,\ldots$.
Programs
(P1)
Sage
import numberdb.sage
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

s = ComplexBallField(400)(QQ(1) / QQ(2))
print(s.zetaderiv(1).real())
Links
Similar tables
Values of the logarithmic derivative $\zeta'(s)/\zeta(s)$ of the Riemann zeta function at rational numbers —   stores $\zeta'(s)/\zeta(s)$; dividing this table's values by the corresponding values of $\zeta(s)$ gives that table where $\zeta(s)\neq0$, but a reader holding the quotient is likely looking for the logarithmic derivative table
Values of the Riemann zeta function at rational numbers —   stores $\zeta(s)$ at the same kind of rational arguments, whose derivative in $s$ is recorded here
Bendersky constants $A_k$ —   stores the displayed constants $A_k$ for $k\neq1$ that give closed forms for $\zeta'(-k)$; the classical $k=1$ member is in the table of the Glaisher-Kinkelin constant
Taylor coefficients of the completed Riemann zeta function at 1/2 —   stores derivatives of the completed zeta function at the centre of the critical strip
Values of the Hurwitz zeta function at pairs of rational numbers —   $\zeta(s)$ is the Hurwitz zeta function at $x=1$
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

Each value is computed as a Sage complex ball with arb at numberdb.bits(digits, losing=96) bits. The imaginary part is required to contain zero before the real part is returned.

more

The entries are checked against the values of $\zeta(s)$ in the zeta table, the identities in (1) and (2), and independent mpmath computations at selected half-integer arguments.

Table is complete: no (it holds every rational argument $s=a/b$ in lowest terms with $b\leq4$ and $-20\leq s\leq20$ at which $\zeta'(s)$ is finite)


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