Show, $\sum_{\chi\neq\chi_0}(\sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\chi(a)(\frac{b^2-a^2}{p})(\frac{b^2-1}{p}))^2|L(1,\chi)|=O(p^{7/2}\log p)$.

Question

I can show, it is bounded by $p^4\log p$. But I think the bound should be smaller.