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](https://mast.queensu.ca/%7Emurty/class-groups_final_revised.pdf)) 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.



Calculating the explicit constant – Siegel zeros and class numbers
One place to find this worked out in detail is the paper ["On the Siegel-Tatuzawa theorem"](https://www.impan.pl/shop/en/publication/transaction/download/product/102670) by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a quadratic Dirichlet character with conductor $d > 10^{6}$ and if $L(s,\chi)$ is nonzero on $(\beta,1)$ and $1-\beta$ is small (specifically $(1 - \beta)^{-1} < 11.657 \log(d)$), then
$$
 L(1,\chi) > 1.507 (1 - \beta).
$$
Also given in Lemma 1 is a lower bound on $L(1,\chi)$ under the assumption that $L(s,\chi)$ doesn't vanish on $(0,1)$.


Combining this with the Dirichlet class number formula $L(1,\chi) = \frac{2 \pi h(d)}{w\_{d} \sqrt{d}}$ gives the result you seek with $c\_{1} = \frac{1.507}{\pi} c$ provided $c < \frac{1}{11.657}$ and $d > 4$.


