A075700 -id:A075700 - cp's OEIS frontend

A375366 Decimal expansion of (1 + log(Pi/2))/Pi.

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

Offset: 0

R. J. Mathar, Aug 13 2024

			0.46205312570704533519089882494319344697793658184986...

Cf. A000796, A075700, A375367.

Maple
```
(1+log(Pi/2))/Pi ; evalf(%) ;
```

RealDigits[(Log[Pi / 2] + 1)/ Pi, 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

(1+log(Pi/2))/Pi \\ Stefano Spezia, Aug 23 2025

Equals Integral_{x=0..1} sin(Pi*x) log(Gamma(x)) dx.

A334851 Decimal expansion of the number x such that 1 = Integral_{0..x} Log(gamma(t)) dt.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 27 2020

			x = 2.75501685669048486879290550748147369075005975463704644144...

Cf. A075700.

x /. FindRoot[x Log[Gamma[x]] - x LogGamma[x] + PolyGamma[-2, x] - 1, {x, 3},  WorkingPrecision -> 120]  (* Peter J. C. Moses, Jun 27 2020 *)

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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A375366 Decimal expansion of (1 + log(Pi/2))/Pi.

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A334851 Decimal expansion of the number x such that 1 = Integral_{0..x} Log(gamma(t)) dt.

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

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