OFFSET
0,1
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
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.
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
EXAMPLE
0.9497031262940093952634984917457415158736519509096929448809176543683...
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]
CROSSREFS
KEYWORD
AUTHOR
Vaclav Kotesovec, May 29 2015
STATUS
approved