Numerical verification of Littlewood's bounds for |L(1,χ)|

Let $L(s,chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $chi$ defined $mod q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $ert L(1,chi) ert$ using the values of Euler's $Gamma$ function. We also introduce an alternative way of computing $log Gamma(x)$ and $psi(x)= Gamma^prime/Gamma(x)$, $xin(0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $ert L(1,chi) ert$, where $chi$ runs over the non trivial primitive Dirichlet characters $mod q$, for every odd prime $q$ up to $10^7$.