A247508 - OEIS

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.

%I #11 Apr 05 2024 10:42:25

%S 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,

%T 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,

%U 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

%N 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.

%H J. M. Borwein, I. J. Zucker and J. Boersma, <a href="http://carma.newcastle.edu.au/MZVs/mzv-week05.pdf">The evaluation of character Euler double sums</a>, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 15 l_2.

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

%e -2.0335765060720546009120689697005182499923760756130461855 ...

%p 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

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

%Y Cf. A244839, A247450.

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Sep 18 2014