OFFSET
1,1
LINKS
Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.
FORMULA
-Integral_{0..Pi/3} log(2*sin(x/2))^3 dx = (1/2)*Pi*zeta(3) + (9/4)*im( PolyLog(4, (-1)^(1/3)) - PolyLog(4, -(-1)^(2/3))).
Also equals 6 * 5F4(1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2; 1/4) (with 5F4 the hypergeometric function).
EXAMPLE
6.00949754981888891620478870620327074059696329743956841883606392675151...
MATHEMATICA
RealDigits[(1/2)*Pi*Zeta[3] + (9/4)*Im[ PolyLog[4, (-1)^(1/3)] - PolyLog[4, -(-1)^(2/3)]], 10, 105] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jun 09 2015
STATUS
approved