Let's define $N(\alpha,T,\chi)=\sharp\lbrace \rho=\sigma+i\gamma: L(\rho,\chi)=0, \alpha\leq \sigma<1, |\gamma|\leq T\rbrace$ , where $\chi$ is a primitive Dirichlet character. We know, from Gallagher's paper, that

$$\sum_{q\leq T}\sum_{\chi\pmod{q}}N(\alpha,T,\chi)\ll T^{c(1-\alpha)}$$

for every $0\leq \alpha<1$ with $c>0$ an absolute costant.

But what about an explicit value of such costant $c$? I need some good and explicit estimate of the sum above. Can anyone help me?