A258763 -id:A258763 - cp's OEIS frontend

A258759 Decimal expansion of Ls_3(Pi/3), the value of the 3rd basic generalized log-sine integral at Pi/3 (negated).

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

Offset: 1

Jean-François Alcover, Jun 09 2015

			-2.0096660811305439002623537543491645038479353700110717949908496919...

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

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)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[-7*Pi^3/108, 10, 105] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^2 dx = -7*Pi^3/108.

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

Original entry on oeis.org

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

Jean-François Alcover, Jun 09 2015

			6.00949754981888891620478870620327074059696329743956841883606392675151...

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

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)).

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

-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).

A258761 Decimal expansion of Ls_5(Pi/3), the value of the 5th basic generalized log-sine integral at Pi/3 (negated).

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jun 09 2015

			-24.01253312551691461501571396363162679502884841064631502190162...

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

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)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[-24*HypergeometricPFQ[Table[1/2, {6}], Table[3/2, {5}], 1/4], 10, 103] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^4 dx = -1543*Pi^5/19440 + 6*Gl_{4, 1}(Pi/3), where Gl is the multiple Glaisher function.

Also equals -24 * 6F5(1/2,1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2,3/2; 1/4) (with 6F5 the hypergeometric function).

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

Original entry on oeis.org

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

Jean-François Alcover, Jun 09 2015

			120.0207613710553001755048886391927614834489250443014689821689519463 ...

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

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)).

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

-Integral_{0..Pi/3} log(2*sin(x/2))^5 dx = (15/2)*Pi*zeta(5) + (35/36)*Pi^3*zeta(3) - (135/4)*Im(-PolyLog(6, (-1)^(1/3)) + PolyLog(6, -(-1)^(2/3))).

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

cp's OEIS Frontend

A258759 Decimal expansion of Ls_3(Pi/3), the value of the 3rd basic generalized log-sine integral at Pi/3 (negated).

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

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A258761 Decimal expansion of Ls_5(Pi/3), the value of the 5th basic generalized log-sine integral at Pi/3 (negated).

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

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