A266274 -id:A266274 - cp's OEIS frontend

A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

1, 3, 2, 2, 8, 0, 9, 9, 7, 5, 0, 4, 2, 1, 2, 5, 1, 4, 5, 2, 7, 0, 9, 8, 2, 1, 1, 5, 8, 5, 7, 8, 5, 5, 1, 8, 6, 8, 0, 6, 4, 8, 0, 0, 9, 9, 9, 9, 5, 5, 0, 3, 1, 4, 5, 8, 8, 4, 7, 4, 5, 0, 1, 9, 2, 4, 1, 4, 2, 9, 1, 5, 7, 1, 9, 9, 4, 0, 4, 2, 9, 3, 8, 7, 7, 8, 3, 9, 4, 6, 4

Offset: 3

G. C. Greubel, Dec 26 2015

			132.28099750421251452709821158578551868064800999955031458847450192414...

G. C. Greubel, Table of n, a(n) for n = 3..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)).

Mathematica
```
RealDigits[N[Zeta'[-20], 100]]
```

zeta'(-20) = (9280784638125*zeta(21))/(8*Pi^20) = - log(A(20)).

Equals (174611/1320)*(zeta(21)/zeta(20)).

Offset corrected by Rick L. Shepherd, May 30 2016

A271179 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(19) (negated).

Original entry on oeis.org

6, 3, 8, 9, 4, 2, 2, 3, 0, 8, 8, 8, 3, 7, 2, 6, 0, 5, 2, 2, 0, 9, 7, 2, 8, 6, 3, 5, 2, 9, 0, 1, 9, 3, 4, 7, 3, 3, 7, 9, 2, 0, 4, 8, 5, 3, 5, 7, 1, 2, 9, 7, 7, 0, 2, 6, 9, 3, 0, 7, 0, 2, 4, 4, 1, 8, 6, 3, 4, 2, 3, 0, 5, 1, 1, 7, 4, 8, 4, 3, 0, 2, 2, 4, 0, 6, 4, 8, 2, 1, 4, 6, 6, 9, 4, 6

Offset: 2

G. C. Greubel, Apr 01 2016

The logarithm of the nineteenth Bendersky constant.

			-63.894223088837260522097286352901934733...

G. C. Greubel, Table of n, a(n) for n = 2..2003

Cf. A266566, A266274.

RealDigits[(BernoulliB[20]/20)*(EulerGamma + Log[2*Pi] - Zeta'[20]/Zeta[20]), 10, 100][[1]]

log(A(19)) = (1/20)*HarmonicNumber(19)*Bernoulli(20) - RiemannZeta'(-19).

log(A(19)) = (Bernoulli(20)/20)*(EulerGamma + log(2*Pi) - Zeta'(20)/Zeta(20)).

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

3, 6, 0, 8, 5, 4, 3, 3, 9, 5, 9, 9, 9, 4, 7, 6, 0, 7, 3, 4, 7, 4, 2, 0, 8, 0, 6, 3, 6, 3, 9, 5, 1, 0, 6, 5, 8, 8, 4, 8, 5, 2, 7, 8, 7, 9, 1, 8, 6, 3, 2, 2, 1, 0, 8, 1, 4, 3, 7, 6, 2, 8, 1, 2, 7, 5, 8, 0, 8, 1, 0, 6, 1, 2, 6, 6, 5, 6, 5, 1, 0, 3, 0, 9, 5, 7, 3, 3, 0, 8, 5, 0, 8, 3, 0, 9, 1, 6, 0, 2, 8, 5, 0, 8, 1

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

cp's OEIS Frontend

A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

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A271179 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(19) (negated).

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A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

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PARI