A258754 Decimal expansion of Ls_8(Pi), the value of the 8th basic generalized log-sine integral at Pi.

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

Offset: 4

Author

Jean-François Alcover, Jun 09 2015

Keywords

• nonn
• cons
• easy

Examples

			5040.03987911504516434562143833539315930537596167748200200213853916...
		

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)).

Programs

Mathematica

RealDigits[(2835/4)*Pi*Zeta[7] + (315/8)*Pi^3*Zeta[5] + (133/32)*Pi^5*Zeta[3], 10, 105] // First

PARI

-intnum(t=0,Pi,log(2*sin(t/2))^7) \\ Hugo Pfoertner, Jul 22 2020

Formula

-Integral_{t=0..Pi} log(2*sin(t/2))^7 = (2835/4)*Pi*zeta(7) + (315/8)*Pi^3*zeta(5) + (133/32)*Pi^5*zeta(3).

Also equals 7th derivative of -Pi*binomial(x, x/2) at x=0.