A081855 Decimal expansion of Gamma''(1).

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

Offset: 1

Author

Benoit Cloitre, Apr 11 2003

Keywords

• cons
• nonn

Comments

Also the decimal expansion of the Integral_{x>=0} exp(-x)*(log(x))^2 dx. - Robert G. Wilson v, Aug 18 2017

Examples

			1.978111990655945110790791303001269415878367... [corrected by _Georg Fischer_, Jul 29 2021]
		

References

• Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 179.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.2, p. 31.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.

Crossrefs

Cf. A001620, A013661, A155969, A261509.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L,2); // G. C. Greubel, Aug 29 2018

Mathematica

EulerGamma^2 + Zeta[2] // RealDigits[#, 10, 105] & // First (* Jean-François Alcover, Apr 29 2013 *)
RealDigits[ Integrate[ Exp[-x]*Log[x]^2, {x, 0, Infinity}], 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2017 *)

PARI

Euler^2+zeta(2) \\ Charles R Greathouse IV, Aug 18 2017

PARI

intnum(x=0,[oo,1],exp(-x)*log(x)^2) \\ Charles R Greathouse IV, Aug 18 2017

Formula

The second derivative of Gamma(x) at x=1 is Gamma^2+zeta(2) = 1.97811199... where Gamma is the Euler constant and zeta(2) = Pi^2/6.