A247508 Decimal expansion of L_2 = -Integral_{x=0..Pi/2} log(2*sin(x/2))^2 dx, a constant appearing in the evaluation of Euler double sums not expressible in terms of well-known constants.

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

Offset

1,1

Links

Table of n, a(n) for n=1..100.

J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 15 l_2.

Formula

L_2 = -Pi^3/24 - (1/2)*Sum_{m >= 1} (Sum_{n=1..m} ((-1)^(n-1)/(2*n-1))/m^2).

Example

-2.0335765060720546009120689697005182499923760756130461855 ...

Maple

evalf(-Pi^3/24 - (1/2)*sum(sum((-1)^(n-1)/(2*n-1)/m^2, n=1..m), m=1..infinity), 100) # Vaclav Kotesovec, Sep 18 2014

Mathematica

digits = 100; L2 = -NIntegrate[Log[2*Sin[t/2]]^2, {t, 0, Pi/2}, WorkingPrecision -> digits+10]; RealDigits[L2, 10, digits] // First

Crossrefs

Cf. A244839, A247450.

Sequence in context: A202035 A369494 A343762 * A112239 A298814 A343980

Adjacent sequences: A247505 A247506 A247507 * A247509 A247510 A247511

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 18 2014

Status

approved