An explicit upper bound for $L(1,\chi )$ when $\chi$ is quadratic

Abstract

We consider Dirichlet $L$-functions $L(s,\chi )$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1,\chi )|\le \frac{1}{2}\log q$ for all $q\ge 2\cdot 10^{23}$ and $|L(1,\chi )|\le \frac{9}{20}\log q$ for all $q\ge 5\cdot 10^{50}$.

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