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Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principalprimitive Dirichlet character $\chi$ defined modulo $q$, where $q\ge 3$. Weprove, under the assumption of the Generalised Riemann Hypothesis, the validityof estimates given by Lamzouri, Li, and Soundararajan on $\vert L(1,\chi)\vert$. As a corollary, we have that similar estimates hold for the classnumber of the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, $q\ge 5$.

Uniform effective estimates for $\vert L(1,\chi)\vert$