A256358 -id:A256358 - cp's OEIS frontend

A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

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

Offset: 0

Ilya Gutkovskiy, Nov 11 2021

First derivative of the Dirichlet eta function at 3.

			0.0597059061601953583634292662879256783169268731565...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A002117, A091812, A197070, A210593, A244115, A256358.

Flatten[{0, RealDigits[(Log[2] Zeta[3] + 3 Zeta'[3])/4, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^3) \\ Michel Marcus, Nov 11 2021

Equals (log(2) * zeta(3) + 3 * zeta'(3)) / 4.

A271533 Decimal expansion of the derivative of the Dirichlet function eta(z) at z = -1.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 09 2016

This entry completes the values of the derivatives eta'(z) at z = 0,1,i,-1,-i (see crossrefs).

			0.265214370914704351169348273575616405600275762885520266292673582574...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A074962, A256358 (eta'(0)), A091812 (eta'(1)), A271525 (real(eta'(i))), A271526 (-imag(eta'(i))) .

RealDigits[3*Log[Glaisher] - Log[2]/3 - 1/4, 10, 120][[1]] (* G. C. Greubel, Apr 09 2016 *)
RealDigits[DirichletEta'[-1], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)

\\ Derivative of Dirichlet eta function (fails for z=1):
derdireta(z)=2^(1-z)*log(2)*zeta(z)+(1-2^(1-z))*zeta'(z);
derdireta(-1) \\ Evaluation

eta'(-1) = 3*log(A) - log(2)/3 - 1/4, where A = A074962 is the Glaisher-Kinkelin constant.

A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

First derivative of the Dirichlet eta function at 4.

			0.0334788045785650663859568547887377997137597304057...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A013662, A091812, A210593, A256358, A261506, A267315, A349220.

Flatten[{0, RealDigits[(Pi^4 Log[2] + 630 Zeta'[4])/720, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^4) \\ Michel Marcus, Nov 12 2021

Equals (Pi^4 * log(2) + 630 * zeta'(4)) / 720.

cp's OEIS Frontend

A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

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A271533 Decimal expansion of the derivative of the Dirichlet function eta(z) at z = -1.

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A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

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