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Values of the Hurwitz zeta function at pairs of rational numbers
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Numbers
$x$
$s$ 
$\zeta(s,x)$
1/2
0:
0
1/2
1/2:
-0.604898643421630370247265914235955499759762545130247380378546648082187253495060357327403956918349554
1/2
-1:
1/24
1/2
2:
4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276115
1/2
-2:
0
1/2
3:
8.414398322117159997798167130580149935354904046383492172545900887392867440504191631305191115265346807
1/2
-3:
-7/960
1/2
4:
16.23484850566707287274005544811751854162126427878090361524464323166180924280453269835364453912549875
1/2
-4:
0
1/2
5:
32.14476040944446771627233008016805920976950850455929717119997301501791128337472602136133630327113342
1/2
-5:
31/8064
1/2
6:
64.09261290502029580201462957682799325781450187206977439607174583227292752282334756661476061012771574
1/2
-6:
0
1/2
7:
128.0603582275041990086542888309241884691826721917853157149949583445933877463041783889543150520591181
1/2
-7:
-127/30720
1/2
8:
256.0397258304758065415647358197063786410350016157117551839231016764086529065609476707982437449548526
1/2
-8:
0
1/2
9:
512.0262887341280115675227650777625629081445900627881545963328707299001622824220418636508583239836585
1/2
-9:
511/67584
1/2
10:
1024.017450355757901299900315955026354397157980790460500154718141512926168850406952348173933888424522
1/2
-10:
0
1/3
0:
1/6
1/3
1/2:
-0.118083327934221719094454170794102733145883537707991498915725290287637208406276710158079046042925204
1/3
-1:
1/36
1/3
2:
10.09559712542709408179200409989251636051890411928097814194168320083102131254508547675242867122397919
1/3
-2:
-1/81
1/3
3:
27.56106119970080377622787797740750928454209531301488108286446031855695818017193319714478144925225572
1/3
-3:
-13/3240
1/3
4:
81.36396942396904029262846038701759877256406348918475617709462160986569294762753996597580083920752837
1/3
-4:
1/243
1/3
5:
243.2552899644188750622367864472924936516933560790675043589542498693177675224814831504573116405904792
1/3
-5:
121/61236
1/3
6:
729.1851269683050198294046987264106157086812142094442012396994831697000326571877903398898562832139889
1/3
-6:
-7/2187
1/3
7:
2187.136405211307613421908035527481041164135476102894927623721421707753520184137845291494795761880309
1/3
-7:
-1093/524880
1/3
8:
6561.101326791542015765966785009244563311871278246751390764199152186151444939797401715431025774482362
1/3
-8:
809/177147
1/3
9:
19683.07559436060070226411091341654961910071301324986541286022285514733842001902645914690201414458519
1/3
-9:
9841/2598156
1/3
10:
59049.05652895576589396163505673037324116931135174361071569061828009063702321261775229521634778654452
1/3
-10:
-1847/177147
2/3
0:
-1/6
2/3
1/2:
-0.950970369576687437180062727168834864862466022701437925449407172954472452266984832316328472300344077
2/3
-1:
1/36
2/3
2:
3.063875409358717409987317233275685153232695090373409359942782634129038450680521513916602533734090446
2/3
-2:
1/81
2/3
3:
3.692418282448647644165314221890190475347548287838089843734600120330835170272207147703071264590460987
2/3
-3:
-13/3240
2/3
4:
5.221889272922015028651835336275833449416012664313396437543475625663956347329967758576970036128465000
2/3
-4:
-1/243
2/3
5:
7.681226780276647109953661275309775018120226440395396138800378183402701205798636113073120146235788184
2/3
-5:
121/61236
2/3
6:
11.44062215637395388276435416137952860384191853447319178157402422545379649543755931876959965603961520
2/3
-6:
7/2187
2/3
7:
17.11511514557568604988940844417467532116626729271688460445951483776731141760179217507317828379868474
2/3
-7:
-1093/524880
2/3
8:
25.64612986697285055820837960751560878691150841626474259476377721479272395055482071765320546827776828
2/3
-8:
-809/177147
2/3
9:
38.45359324234944190806729061578455537698135390433509352925573918905100080803934899473437241636660413
2/3
-9:
9841/2598156
2/3
10:
57.67114319163640902615952441566407500212994807565772334730627519310141107753837894131781471950749412
2/3
-10:
1847/177147
1/4
0:
1/4
1/4
1/2:
0.239963524495630955337574310160577225897864436801770042667628937450089970799560852126575361457251515
1/4
-1:
1/96
1/4
2:
17.19732915450711073927131911933522402150689440149416770054533433319414898062924339883662550712747308
1/4
-2:
-1/64
1/4
3:
64.66386996876846016666898358942199494364490475141907638432814165337786467948247256278746549366427342
1/4
-3:
-7/15360
1/4
4:
256.4636906681980660037985406381725094614003077742771695020521732630884188637825907275536963449783815
1/4
-4:
5/1024
1/4
5:
1024.348974526580572231592798021748291490146647119740405661686974384011567423352323818087214974846666
1/4
-5:
31/516096
1/4
6:
4096.270948064010767048866559996198797991354650373334295229946838751185397971242793336499853459200729
1/4
-6:
-61/16384
1/4
7:
16384.21345519957167514794542195223925078161906440846189589337973562603080264837073854843660815358773
1/4
-7:
-127/7864320
1/4
8:
65536.16938670901741620606758211393365745829177159850765229054509593862012080375191803325135946012871
1/4
-8:
1385/262144
1/4
9:
262144.1349217240825743509781002783992122858405081048305721835993467866193378546960827828761948791914
1/4
-9:
511/69206016
1/4
10:
1.048576107683114753926745099997368583807477957060385151250594582192326177389373061288826576240448215e6
1/4
-10:
-50521/4194304
3/4
0:
-1/4
3/4
1/2:
-1.095419389883587397979807548439919738605331222684104152831968377193433246779001120075590780002258780
3/4
-1:
1/96
3/4
2:
2.541879647671606498397662880417078249120504412987413552281364419245940664209167087166921300309631381
3/4
-2:
1/64
3/4
3:
2.651316608168819815716353455219204539194327619648860996039065445765074844551060487654063428458501035
3/4
-3:
-7/15360
3/4
4:
3.293885422475099960042346531707787204539920686217288341862118443500529021089932446104616281029598543
3/4
-4:
-5/1024
3/4
5:
4.283358575642394689121764543629603222477625026157103816712162096561593644638908865475546729829603337
3/4
-5:
31/516096
3/4
6:
5.656277857288164280069732920792770508773469439131266118644894514281963489451450926844825588973077811
3/4
-6:
61/16384
3/4
7:
7.512397920965797959803548406056873273762976140058515625974932481922828878564095237715718509979389233
3/4
-7:
-127/7864320
3/4
8:
10.00042589278905843450478773089927464666864202370167479376893322199502327585068569109903924831356489
3/4
-8:
-1385/262144
3/4
9:
13.32491014945934822067761953603299668418960404270458113883046692226375074538935140636326700044167789
3/4
-9:
511/69206016
3/4
10:
17.76148118133700435282354057840309521181526904640090783679471691021951344365791570353206129849587431
3/4
-10:
50521/4194304
1/5
0:
3/10
1/5
1/2:
0.532174445889806671755534973682753163907785372688433310392490811569029588536463186427013369377006992
1/5
-1:
-1/300
1/5
2:
26.26737720542377912330246621503190788838988399080040938445663873863547305090742040997292929530322331
1/5
-2:
-2/125
1/5
3:
125.7390180572179665295990450183751810318670140080030728087917456338750718509241827534676472634140045
1/5
-3:
29/15000
1/5
4:
625.5408324774542966919937573694648175518266362163601873959594684219451756031007481803297908479664132
1/5
-4:
74/15625
1/5
5:
3125.425520748860842139697326202328066131604329611228222410317162446036094974051555959611068691983475
1/5
-5:
-4537/3937500
1/5
6:
15625.34491398681477095949518389630450910838137410614560763486851789108982014592330292412254650716668
1/5
-6:
-278/78125
1/5
7:
78125.28343945023873232532559418550167233249467099440300762281115254535536264672969959076853092252969
1/5
-7:
23789/18750000
1/5
8:
390625.2344941637267702957200471310783276487969835899003670134914979076237239819387419760776698860561
1/5
-8:
9826/1953125
1/5
9:
1.953125194666358282328038433493928263415015939415316239943147822296141735735913645160795901123882154e6
1/5
-9:
-15034541/6445312500
1/5
10:
9.765625161891642127719095943452167829820539578605226365169695885567770681715956085648576027568307078e6
1/5
-10:
-559526/48828125
2/5
0:
1/10
2/5
1/2:
-0.338253094704176860308446788794021337302279787188541222813123493038699840685444046691641052126198010
2/5
-1:
11/300
2/5
2:
7.275356590529597437537027549009956633947223294273610399702948252591549141138142005773652939442825857
2/5
-2:
-1/125
2/5
3:
16.11956431178917847526563373333078988139648151118044162244746165123216647793720447163213288797295065
2/5
-3:
-91/15000
2/5
4:
39.36587650565784908039775595817582883474593740138412624029024040012626104336992183122691050678510939
2/5
-4:
43/15625
2/5
5:
97.85796972866625627771007425557579418311558842525579166114181711188350416758333219695897956608696317
2/5
-5:
12347/3937500
2/5
6:
244.2795205771564119835815291193307240746833608159617907912985502849020157497607815043744798738565136
2/5
-6:
-169/78125
2/5
7:
610.4488402761203393342502241494892325983465217745482789211648773363179450197583479557097363153320351
2/5
-7:
-62851/18750000
2/5
8:
1525.947640117580219726525174009511321291983016416641818979554528237311948901861736764330091465724601
2/5
-8:
6047/1953125
2/5
9:
3814.746062834144376141616942002989074826972698932998777512750473947787507823796234907348126186345873
2/5
-9:
39448591/6445312500
2/5
10:
9536.777898666596874548990813622451851459818049191352822783630556970057791560271277006130380284121939
2/5
-10:
-345433/48828125
3/5
0:
-1/10
3/5
1/2:
-0.823401063598787756167436211412559211436591352099019193494026522648888474338718786401079672905666199
3/5
-1:
11/300
3/5
2:
3.636209670902357675994414753710295290927094622046135053847282972777570448220480884187711787862108550
3/5
-2:
1/125
3/5
3:
4.981415768571729079787502059245503474882889555367658878004468568641562692521117861901821480886473638
3/5
-3:
-91/15000
3/5
4:
7.901173507557629232949738279653828547807077121950667571998877743184762553552445535721032953983785256
3/5
-4:
-43/15625
3/5
5:
12.96636383038591045695417371555913860734416996982852797760968167412752123534171995793488113052221801
3/5
-5:
12347/3937500
3/5
6:
21.49693256473723858930765933100398148344356326200081447328617078879034279895392207873530664357277352
3/5
-6:
169/78125
3/5
7:
35.76110823383758590763019160412171109236002707390163591619186241403714515333416569149126782681831625
3/5
-7:
-62851/18750000
3/5
8:
59.56122190301949505942116905656409493354634482407120772444940159960588737839534803163760148920441006
3/5
-8:
-6047/1953125
3/5
9:
99.24377739741052680554505224913172085670275451948961774091638482704594705186960837926239157174343531
3/5
-9:
39448591/6445312500
3/5
10:
165.3908856767040688928449169622095687065576104540665647721408525701410120356560322012118654033117015
3/5
-10:
345433/48828125
4/5
0:
-3/10
4/5
1/2:
-1.175617731723806841901564042586502651424715631857092474774260107728303901331851225565691581631706418
4/5
-1:
-1/300
4/5
2:
2.299474137501700238504055481752444727990595721843007667646527540875586649410777672072799592266051203
4/5
-2:
2/125
4/5
3:
2.215057854210817304915351216468324466711915175670688032997997008639136496120318096118926695284143190
4/5
-3:
29/15000
4/5
4:
2.561815345080456500645055034394296397064943257590609185928571871875065300645444704233878518885440435
4/5
-4:
-74/15625
4/5
5:
3.112452759974640984824205518311742088256704517663082558309264539435294115366749651069735495891886266
4/5
-5:
-4537/3937500
4/5
6:
3.846633316324937367243762706703113271488166089195837326339727438903846965552190013228298287077529459
4/5
-6:
278/78125
4/5
7:
4.785558225142266465137774526409430956539585755855777683659869224083439245277254940510064361040783829
4/5
-7:
23789/18750000
4/5
8:
5.969831281483140377876217006706845441973541976400947320848373218021706296485047572633714136443726196
4/5
-8:
-9826/1953125
4/5
9:
7.455723459161721668676566105189013188828031021979406819727697266453427005734187505001104293726762037
4/5
-9:
-15034541/6445312500
4/5
10:
9.316062037933139942888557216508889686527649947405435826735978728322563132780412875207854834267238639
4/5
-10:
559526/48828125
Definition
The list contains values $\zeta(s,x)$ of the Hurwitz zeta function at rational numbers $s$ and $x$.
Parameters
$x$
—   rational number ($x > 0$)
$s$
—   rational number ($s \neq 1$)
Comments
(1)
The Hurwitz zeta function $\zeta(s,x)$ is the meromorphic continuation (with respect to $s$) of the series $\sum_{n=0}^\infty (n+x)^{-s}$, $\Re(s) > 1$, $\Re(x) > 0$.
(2)
At $s=1$, $\zeta(s,x)$ has a simple pole (for any fixed $x$).
(3)
$\zeta(s,1)$ is the Riemann zeta function.
(4)
At rational shifts $x=r/q$, Hurwitz values are the residue-class components of Dirichlet $L$-values [2]. Nonpositive integer orders are Bernoulli-polynomial values, positive integer orders include the polygamma special values [3], and $s=1/2$ gives the components of central Dirichlet $L$-values. The rational shift is therefore an arithmetic parameter; this does not justify a separate continuous rational mesh in the order $s$.
Programs
(P1)
Python
from values import SelectedValues
from numberdb._write import to_text

generator = SelectedValues('T94')
for params in generator.enumerate():
    print(params, to_text(generator.value(params, 100), 100))
Links
Data properties
Entries are of type: real number
Table is complete: no (it holds every reduced shift $0<x<1$ of denominator at most $5$, at integer orders $-10\leq s\leq10$, $s\ne1$, and at $s=1/2$; these are finite conductor and integer-order reference bounds for the arithmetic families in (4); the $x=1$ group refers to the Riemann zeta table)
How well the digits are known: proven
How they were obtained:

Every selected entry is checked against a fresh exact rational result or an Arb enclosure at 493 working bits (100 decimal digits plus 160 guard bits). Rational arguments and square roots are enclosed before evaluation, so their uncertainty is propagated.

more

Existing decimal strings and surviving row annotations are retained only after the recomputed enclosure is strictly contained in the interval their last digit denotes. New inexact entries are written to 100 significant digits; proved rational values remain exact. Every inexact ball is finite and has at least 488 relative accuracy bits. All 189 entries also pass mpmath checks at 150 and 200 decimal digits; 21 entries have additional closed-form or functional-identity checks. The mpmath agreements supplement, and do not establish, the rigorous enclosure. A nonzero control and ten-last-place perturbation controls pass. The attached checks-evidence.json records this selection, and checks.py reproduces the checks without writing to the website.



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