A261508 -id:A261508 - cp's OEIS frontend

A257549 Decimal expansion of zeta''(0) (negated).

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

Offset: 1

Jean-François Alcover, Apr 29 2015

Essentially the same as A245273. - R. J. Mathar, Apr 30 2015

			2.00635645590858485121010002672996043819899491016091988116986828...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.
Richard E. Crandall, Unified algorithms for polylogarithm, L-series, and zeta variants. p. 15.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 104.
Eric Weisstein's World of Mathematics, Riemann Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.

Cf. A075700, A082633, A261508, A201994.

evalf(-Zeta(2, 0), 120); # Vaclav Kotesovec, Apr 29 2015

RealDigits[ StieltjesGamma[1] + EulerGamma^2/2 - Pi^2/24 - (1/2)*(Log[2] + Log[Pi])^2, 10, 104] // First

-zeta''(0) \\ Charles R Greathouse IV, Mar 10 2016

zeta''(0) = gamma_1 + gamma^2/2 - Pi^2/24 - (1/2)*(log(2)+log(Pi))^2, where gamma_1 is the first Stieltjes constant.

A385612 Decimal expansion zeta''''(0) (negated).

Original entry on oeis.org

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

Offset: 2

Artur Jasinski, Jul 04 2025

n-th derivative of zeta function at 0 is close to -n!, which here is the present constant close to 4! = 24.

			23.997103188013707958987219527741...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.

Cf. A075700, A257549, A261508.

Cf. A001620, A061444, A082633, A086279, A086280.

evalf(-Zeta(4, 0), 120); # Vaclav Kotesovec, Jul 04 2025

RealDigits[-3 EulerGamma^4/2 - EulerGamma^2 Pi^2/4 + 19 Pi^4/480 - 4 EulerGamma^3 Log[2 Pi] - 3 EulerGamma^2 Log[2Pi]^2 +  Pi^2 Log[2 Pi]^2/4 + Log[2 Pi]^4/2 - 6 EulerGamma^2 StieltjesGamma[1] - Pi^2 StieltjesGamma[1]/2 - 12 EulerGamma Log[2 Pi] StieltjesGamma[1] - 6 Log[2 Pi]^2 StieltjesGamma[1] - 6 EulerGamma StieltjesGamma[2] - 6 Log[2Pi] StieltjesGamma[2] - 2 StieltjesGamma[3] + 4 Log[2 Pi] Zeta[3],10,105][[1]]

PARI
```
-zeta''''(0)
```

Equals -3*gamma^4/2 - gamma^2*Pi^2/4 + 19*Pi^4/480 - 4*gamma^3*log(2*Pi) -3*gamma^2*log(2*Pi)^2 + Pi^2*log(2*Pi)^2/4 + log(2*Pi)^4/2 - 6*gamma^2*StieltjesGamma(1) - Pi^2*StieltjesGamma(1)/2 - 12*gamma*log(2*Pi)* StieltjesGamma(1) - 6*log(2*Pi)^2*StieltjesGamma(1) - 6*gamma*StieltjesGamma(2) - 6*log(2*Pi)*StieltjesGamma(2) - 2*StieltjesGamma(3) + 4*log(2*Pi)*zeta(3).

cp's OEIS Frontend

A257549 Decimal expansion of zeta''(0) (negated).

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