This is problem 8 of Chapter 16 in the book A Classical Approach to Modern Number Theory by Ireland and Rosen.
Let $g(\chi)$ be the classical Gauss sum : $\sum_{x = 1}^{p-1} \chi(x) \zeta^{x} $, $\chi$ the Legendre symbol. $\zeta = e^{\frac{2 \pi i}{p} }$.
Define $P = \prod (1 - \zeta ^n)(1 - \zeta ^{r})^{-1}$ where $r, n$ run respectively over the nonsquares and squares mod p.
Show $$P = e^{g(\chi) L(1, \chi)} $$
where $L(s, \chi)$ is the Dirichlet $L$ function for the unique non-trivial Quadratic character mod $p$ (a prime).
This exercise can be used to compute $L$ explcitly at $1$.
I am trying to express $g(\chi) L (1, \chi)$ in terms of logarithms but I unable to.