Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\log |q|} < \sigma < 1$ for some constant $c$.

I have read in many places (see for example the second page of the article by Ajit Bhand and Ram Murty available here) that the non-existence of Siegel zeros for $ L(s,\chi) $ can be used to prove the bound $$ h(d) > c_1 \frac{\sqrt{d}}{\log(d)}, $$ where $ h(d)$ is the class number of the associated imaginary quadratic field and $c_1$ is another effective constant which can be calculated depending on $c$.

My Question : How to explicitly compute $c_1$ from $c$?

If anyone can direct me to a proof of the above statement, I think that would also suffice.

Thank you, Krishnarjun