A258749 Decimal expansion of Ls_3(Pi), the value of the 3rd basic generalized log-sine integral at Pi (negated).

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Jun 09 2015

Keywords

• nonn
• cons
• easy

Examples

			-2.5838563900249850146230262555917829335187740471570923078453781753171...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.
• Index entries for transcendental numbers.

Crossrefs

Cf. A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Programs

Magma

R:= RealField(100); -Pi(R)^3/12; // G. C. Greubel, Aug 23 2018

Mathematica

RealDigits[-Pi^3/12, 10, 103] // First

PARI

-Pi^3/12 \\ G. C. Greubel, Aug 23 2018

Formula

-Integral_{0..Pi} log(2*sin(t/2))^2 dx = -Pi^3/12.

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

It can be noticed that Ls_2(Pi) is 0, and that Ls_2(Pi/2) is Catalan's constant 0.915966... (A006752).