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Decimal expansion of Ls_5(Pi), the value of the 5th basic generalized log-sine integral at Pi (negated).
10

%I #10 Nov 26 2024 13:26:17

%S 2,4,2,2,6,5,5,8,3,7,8,8,3,4,7,8,1,7,1,6,6,3,3,6,8,7,0,4,5,1,0,5,3,1,

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

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

%N Decimal expansion of Ls_5(Pi), the value of the 5th basic generalized log-sine integral at Pi (negated).

%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))^4 dx = -19*Pi^5/240.

%F Also equals 4th derivative of -Pi*binomial(x, x/2) at x=0.

%e -24.22655837883478171663368704510531884635713927472260341881815179...

%t RealDigits[-19*Pi^5/240, 10, 105] // First

%o (PARI) -19*Pi^5/240 \\ _Charles R Greathouse IV_, Nov 26 2024

%Y Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

%K nonn,cons,easy

%O 2,1

%A _Jean-François Alcover_, Jun 09 2015