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%I #15 Apr 06 2024 13:44:41
%S 2,5,8,3,8,5,6,3,9,0,0,2,4,9,8,5,0,1,4,6,2,3,0,2,6,2,5,5,5,9,1,7,8,2,
%T 9,3,3,5,1,8,7,7,4,0,4,7,1,5,7,0,9,2,3,0,7,8,4,5,3,7,8,1,7,5,3,1,7,1,
%U 9,9,5,7,6,4,5,5,4,7,5,5,0,3,1,3,0,5,5,8,4,1,9,3,8,3,5,7,3,8,4,9,4,1,9
%N Decimal expansion of Ls_3(Pi), the value of the 3rd basic generalized log-sine integral at Pi (negated).
%H G. C. Greubel, <a href="/A258749/b258749.txt">Table of n, a(n) for n = 1..10000</a>
%H Jonathan M. Borwein, Armin Straub, <a href="https://carmamaths.org/resources/jon/logsin3.pdf">Special Values of Generalized Log-sine Integrals</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F -Integral_{0..Pi} log(2*sin(t/2))^2 dx = -Pi^3/12.
%F Also equals 2nd derivative of -Pi*binomial(x, x/2) at x=0.
%F It can be noticed that Ls_2(Pi) is 0, and that Ls_2(Pi/2) is Catalan's constant 0.915966... (A006752).
%e -2.5838563900249850146230262555917829335187740471570923078453781753171...
%t RealDigits[-Pi^3/12, 10, 103] // First
%o (PARI) -Pi^3/12 \\ _G. C. Greubel_, Aug 23 2018
%o (Magma) R:= RealField(100); -Pi(R)^3/12; // _G. C. Greubel_, Aug 23 2018
%Y Cf. A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Jun 09 2015
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