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%I #8 Apr 06 2024 13:45:36
%S 7,2,0,1,2,4,5,6,8,2,2,6,3,3,1,8,0,1,0,5,3,0,2,9,3,3,1,8,3,5,1,5,6,5,
%T 6,8,9,0,0,6,9,3,5,5,0,2,6,5,8,0,8,8,1,3,8,9,3,0,1,3,7,1,1,6,7,7,8,2,
%U 9,1,8,4,5,9,9,7,3,0,1,2,2,7,2,2,9,5,2,7,7,7,1,1,9,7,8,9,2,3,8,2,3,5,2
%N Decimal expansion of Ls_7(Pi/3), the value of the 7th basic generalized log-sine integral at Pi/3 (negated).
%H Jonathan M. Borwein, Armin Straub, <a href="https://carmamaths.org/resources/jon/logsin3.pdf">Special Values of Generalized Log-sine Integrals</a>.
%F -Integral_{0..Pi/3} log(2*sin(x/2))^5 dx = -74369*Pi^7/326592 - (15/2) * Pi * Zeta[3]^2 + 135*Gl_{6, 1}(Pi/3), where Gl is the multiple Glaisher function.
%F Also equals -720 * 7F6(1/2,1/2,...; 3/2,3/2,...; 1/4) (with 7F6 the hypergeometric function).
%e -720.1245682263318010530293318351565689006935502658088138930137116778...
%t RealDigits[-720*HypergeometricPFQ[Table[1/2, {7}], Table[3/2, {6}], 1/4], 10, 103] // First
%Y Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).
%Y Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)).
%K nonn,cons,easy
%O 3,1
%A _Jean-François Alcover_, Jun 09 2015