Fast Computing Formulas for some Dirichlet L-Series

Abstract

For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $\zeta$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute selected $L-$function values. The new algorithms include $L_k(2)$ for $k = -4$ Catalan's constant, $-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k = 1$ Apery's constant, $5, 8$ and $12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value.