Values of the derivative $\zeta'(s)$ of the Riemann zeta function at rational numbers

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Numbers

$s$

$\zeta'(s)$

-0.9189385332046727417803297364056176398613974736377834128171515404827656959272603976947432986359541976

comment: $-\frac12\log(2\pi)$.

1/2:

-3.922646139209151727471531446714599513730323971506505209568298485254720803150338284880650523104145691

-1:

-0.1654211437004509292139196602427806427640363803352017836665223063573596996665771727595251003325087555

comment: $\frac1{12}-\log A$, where $A$ is the Glaisher-Kinkelin constant [3].

-0.9375482543158437537025740945678649778978602886148299258854334803604438113127075227936894151411515175

comment: $\frac{\pi^2}{6}(\gamma+\log(2\pi)-12\log A)$ [3].

-2:

-0.03044845705839327078025153047115477664700048354497393625297188985903781794493689867779458488087449597

comment: $-\zeta(3)/(4\pi^2)$.

-0.1981262428856368533306818215032857968755427934638350033468899631927256694226511041515708421775957421

-3:

0.005378576357774301144416974210413842895664439742295507059447023223324501997924069586095103870825683267

-0.06891126589612537984882936558744082715001637487137846382758570601842849852762120133478041038984760229

-4:

0.007983811450268624280696670798789303905237693362298876417704739714028740281878657952543961969286982040

comment: $3\zeta(5)/(4\pi^4)$.

-0.02857378050946295008038981708380121264821523555636624823848001794536450116616395297676555989701173090

-5:

-0.0005729859801986352049909941488338745132539872911995212178207918809977350313508312578653993423857005060

-0.01285216513179572507594540145992360882884875391193195136796944988339867117171289588221630843096867868

-6:

-0.005899759143515937450629877408392025579801534620157195865239392206359703759424905723023863007542258385

-0.006033516960875637793965365478530432397640013548691334714307692521376137030039495342180001675585284016

-7:

-0.0007286426801592406524672333546503606119028877209254183186363861541425975455273099130232464416804493796

-0.002901952553710673130400106656218756390516301366166128158831797751028933952751176138154515847293788321

-8:

0.008316161985602247359524426510534214225674122918829999040210532753056917407881235383483452514524403517

-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

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 $s=-2,-4,\ldots$.

(6)

Integer arguments are selected by the special-value identities for $\zeta'$ [4]: the negative even integers are the trivial zeros of $\zeta$, the negative integer derivatives include the Bendersky constants, and the positive even derivatives are related to them by reflection. The centre $s=1/2$ is fixed by the functional equation [5]. The remaining integer orders form an explicitly bounded reference prefix of the integer special-value family, not a selection of all small rational orders.

Programs

(P1)

Python

from values import SelectedValues
from numberdb._write import to_text

generator = SelectedValues('T228')
for params in generator.enumerate():
    print(params, to_text(generator.value(params, 100), 100))

Links

[1]

DLMF 25.2: Definition and Expansions

[2]

DLMF 25.6: Integer Arguments

[3]

DLMF 5.17: Barnes' G-Function (Double Gamma Function)

[4]

DLMF 25.6(ii): Derivatives at Integer Arguments

[5]

DLMF 25.4: Reflection Formulas

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:

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.

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 478 relative accuracy bits. All 41 entries also pass mpmath checks at 150 and 200 decimal digits; 11 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.

Table is complete: no (it holds the integer arguments $-20\leq s\leq20$, $s\ne1$, and the functional-equation centre $s=1/2$; the integer bounds delimit a reference prefix)