A258762 Decimal expansion of Ls_6(Pi/3), the value of the 6th basic generalized log-sine integral at Pi/3.

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

Offset

3,2

Links

Table of n, a(n) for n=3..105.

Jonathan M. Borwein and Armin Straub, Special values of generalized log-sine integrals, ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation, 2011, pp. 43-50; alternative link.

Formula

Equals -Integral_{0..Pi/3} log(2*sin(x/2))^5 dx.

Equals (15/2)*Pi*zeta(5) + (35/36)*Pi^3*zeta(3) - (135/4)*Im(-PolyLog(6, (-1)^(1/3)) + PolyLog(6, -(-1)^(2/3))).

Equals 120 * 7F6(1/2,1/2,...; 3/2,3/2,...; 1/4) (with 7F6 the hypergeometric function).

Example

120.0207613710553001755048886391927614834489250443014689821689519463...

Mathematica

RealDigits[120* HypergeometricPFQ[Table[1/2, {7}], Table[3/2, {6}], 1/4], 10, 103] // First

Prog

(PARI) -intnum(x = 0, Pi/3, log(2*sin(x/2))^5) \\ Amiram Eldar, Jun 29 2026
(PARI) (15/2)*Pi*zeta(5) + (35/36)*Pi^3*zeta(3) + (135/2)*imag(polylog(6, (-1)^(1/3))) \\ Amiram Eldar, Jun 29 2026

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)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258763 (Ls_7(Pi/3)).

Sequence in context: A361015 A028959 A317642 * A079181 A093693 A224447

Adjacent sequences: A258759 A258760 A258761 * A258763 A258764 A258765

Keyword

nonn,cons,easy,changed

Author

Jean-François Alcover, Jun 09 2015

Status

approved