A257549 -id:A257549 - cp's OEIS frontend

A075700 Decimal expansion of -zeta'(0).

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

Offset: 0

Benoit Cloitre, Oct 02 2002

The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - Rick L. Shepherd, Mar 08 2014

For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - Jianing Song, Apr 20 2021

			0.91893853320467274178032...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
Eric Weisstein's World of Mathematics, Log Gamma Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.
Wikipedia, Gamma function.
Wikipedia, Normal curve
Index entries for zeta function.

Cf. A019727, A061444, A257549.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018

evalf(log(2*Pi)/2,120); # Muniru A Asiru, Oct 08 2018

Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)

-zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012

Equals log(2*Pi)/2 = A061444/2 = log(A019727).
Equals Integral_{x=0..1} log(Gamma(x)) dx. - Jean-François Alcover, Apr 29 2013
More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} log(Gamma(x)) dx for any t>=0 (the Raabe formula). - Stanislav Sykora, May 14 2015
Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - Amiram Eldar, Aug 21 2020

Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009

A261508 Decimal expansion of -zeta'''(0).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Aug 22 2015

			6.004711166862254447761060813366375285461807668295980132893...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Stieltjes Constants

Cf. A075700, A257549, A201995, A082633, A086279.

Maple
```
evalf(-Zeta(3, 0), 120);
```

RealDigits[3*Log[2*Pi]*StieltjesGamma[1] + 3*EulerGamma*StieltjesGamma[1] + 3/2*StieltjesGamma[2] - Zeta[3] - 1/2*Log[2*Pi]^3 - 1/8*Pi^2*Log[2*Pi] + 3/2*EulerGamma^2*Log[2*Pi] + EulerGamma^3, 10, 120][[1]]

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

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).

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A075700 Decimal expansion of -zeta'(0).

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A261508 Decimal expansion of -zeta'''(0).

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

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