A258760 Decimal expansion of Ls_4(Pi/3), the value of the 4th basic generalized log-sine integral at Pi/3.

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

Offset: 1

Author

Jean-François Alcover, Jun 09 2015

Keywords

• nonn
• cons
• easy

Examples

			6.00949754981888891620478870620327074059696329743956841883606392675151...
		

Links

• Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Crossrefs

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)).

Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

Programs

Mathematica

RealDigits[(1/2)*Pi*Zeta[3] + (9/4)*Im[ PolyLog[4, (-1)^(1/3)] - PolyLog[4, -(-1)^(2/3)]], 10, 105] // First

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).