A258414 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(24*k)) dx.

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

Offset: 0

Author

Vaclav Kotesovec, May 29 2015

Keywords

• nonn
• cons
• easy

Comments

Integral_{x=-1..1} Product_{k>=1} (1-x^(24*k)) dx = Pi^2/(3*sqrt(3)) = 1.89940625258801878... . - Vaclav Kotesovec, Jun 02 2015

Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=2. - Jianing Song, Nov 16 2019

Examples

			0.9497031262940093952634984917457415158736519509096929448809176543683...
		

Links

• M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
• Vaclav Kotesovec, The integration of q-series.

Crossrefs

Cf. A258232, A258406, A258408.

Cf. A196530, A110161.

Cf. A346451, A346450, A346449.

Programs

Maple

evalf(Pi^2/(6*sqrt(3)), 120);

Mathematica

RealDigits[Pi^2/(6*Sqrt[3]),10,120][[1]]
N[Sum[(-1)^n/(12*n*(3n-1)+1),{n,-Infinity,Infinity}],105]

Formula

Equals Pi^2/(6*sqrt(3)).

Equals Sum_{k>=1} A110161(n)/k^2 = Sum_{k>=1} Kronecker(12,k)/k^2. - Jianing Song, Nov 16 2019

Equals -Integral_{x=0..oo} log(x)/(x^6 + 1) dx. - Amiram Eldar, Aug 12 2020

Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^2 + (-1)^k/(6*k+1)^2 ). - Sean A. Irvine, Jul 18 2021

Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023