Averages of $L(1,\chi)$

Question

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol and $\mu(n)$ be the Moebius function. The series $$ \sum_{\substack{m,n \in \mathbb{N} \\ m,n\equiv 1 \mod{4}}} \frac{\mu^2(mn)}{(mn)^{1+\epsilon}} \left(\frac{m}{n}\right) $$ converges absolutely for all $\epsilon>0$ but I would like to ask whether it converges for $\epsilon=0$.