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
Examples
-2.0335765060720546009120689697005182499923760756130461855 ...
Links
- 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.
Programs
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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
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Mathematica
digits = 100; L2 = -NIntegrate[Log[2*Sin[t/2]]^2, {t, 0, Pi/2}, WorkingPrecision -> digits+10]; RealDigits[L2, 10, digits] // First
Formula
L_2 = -Pi^3/24 - (1/2)*Sum_{m >= 1} (Sum_{n=1..m} ((-1)^(n-1)/(2*n-1))/m^2).