A240966 -id:A240966 - cp's OEIS frontend

A271170 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(3) (negated).

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

Offset: 0

G. C. Greubel, Apr 01 2016

The logarithm of the third Bendersky constant.

			-0.02065635413555207892219475198819162067344221752007...

G. C. Greubel, Table of n, a(n) for n = 0..2000

log(A(b)): A225746 (b=1), (-1) * A240966 (b=2).

Cf. A243263, A259068.

Join[{0}, RealDigits[(BernoulliB[4]/4)*(EulerGamma + Log[2*Pi] - Zeta'[4]/Zeta[4]), 10, 100] // First]

Equals (Bernoulli(4)/4)*(EulerGamma + log(2*Pi) - (Zeta'(4)/Zeta(4))).

log(A(3)) = HarmonicNumber(3)*Bernoulli(4)/4 - Zeta'(-3).

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)
```

A324992 Decimal expansion of zeta'(-1, 1/2).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			0.053829439326894410047908491727299631045539017902590256244899486116455...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 22.

Cf. A074962, A084448, A240966, A261829, A306635.

evalf(Zeta(1,-1,1/2), 120);
evalf(-log(2)/24 - Zeta(1,-1)/2, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 1/2], 10, 120][[1]]
N[With[{k=1}, -BernoulliB[2*k] * Log[2] / 4^k / k - (2^(2*k - 1) - 1) * Zeta'[1 - 2*k] / 2^(2*k - 1)], 120]

zetahurwitz'(-1, 1/2) \\ Michel Marcus, Mar 24 2019

Equals -log(2)/24 - Zeta'(-1)/2 = A261829 - log(2)/24.
Equals -1/24 - log(2)/24 + log(A)/2, where A is the Glaisher-Kinkelin constant A074962.
Equals (log(Pi) - 1 + gamma)/24 - Zeta'(2)/(4*Pi^2), where gamma is the Euler-Mascheroni constant A001620.

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A271170 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(3) (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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A324992 Decimal expansion of zeta'(-1, 1/2).

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