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

TL;DR

This paper develops rapidly converging series for special values of Dirichlet L-functions with quadratic irrational summands, extending previous methods to new cases using lattice sum techniques and the Guillera-Rogers theory.

Contribution

It introduces new geometrically convergent series for specific L(2, (d·)) values beyond previously solvable cases, leveraging lattice sums and the Guillera-Rogers framework.

Findings

Derived series for L_{-56}(2), L_{-68}(2), L_{-87}(2), L_{-111}(2), L_{-116}(2)

Extended Guillera-Rogers methods to new quadratic irrational cases

Achieved faster convergence in evaluating these L-values

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)\frac1{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.