Fast converging irrational series for $L(2,(\frac{d}{\cdot }))$

Abstract

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $\pi$ and special values of Dirichlet $L$-functions $L_d(2)\equiv L(2,(\frac{d}{\cdot })):={\sum }_{k=1}^{\infty }\left(\frac{d}{k}\right)\frac{1}{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $L_{-56}(2)$, $L_{-68}(2)$, $L_{-87}(2)$, $L_{-111}(2)$, and $L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers.

20 pages. Theorem 1.2 expanded