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

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce fast method compute $\vert L(1,\chi) \vert$ using values of Euler's $\Gamma$ function. We also alternative way computing $\log \Gamma(x)$ and $\psi(x)= \Gamma^\prime/\Gamma(x)$,$x\in(0,1)$. Using such algorithms numerically verify classical Littlewood bounds recent Lamzouri-Li-Soundararajan estimates on \vert$, runs over characters for every prime up $10^7$. The programs used results here described are collected at following address \url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.