Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants (or positive prime discriminants) of quadratic number fields. For such a discriminant let $\chi_j(n) = (\frac{D_j}n)$ be the associated Dirichlet character and $$ L(1,\chi_j) = \sum_{n \ge 1} \frac1n \Big(\frac{D_j}{n}\Big) $$ the value of Dirichlet's L-series at $s = 1$. If we assume that about half prime discriminants $D$ have $(D/p) = +1$ and the other half satisfy $(D/p) = -1$, and if we interchange the limits, then the geometric mean of the values of $L(1,\chi_j)$ is given by \begin{align*} \lim_{k \to \infty} \bigg( \prod_{j=1}^k L(1,\chi_j) \bigg)^{1/k} & = \lim_k \prod_p \Big(\frac{p}{p-1}\Big)^{\frac{k}{2k}} \cdot \Big(\frac{p}{p+1}\Big)^{\frac{k}{2k}} \\ &= \prod_p \Big(\frac{p^2}{p^2-1}\Big)^{1/2} = \sqrt{\zeta(2)} = \frac{\pi}{\sqrt{6}}. \end{align*} Has this result due to Scholz been studied anywhere?

Scholz also believed that if $S$ denotes the set of all fundamental discriminants, then the corresponding limit is equal to $$ \prod \Big( \frac{p^2}{p^2-1} \Big)^{\frac{p}{2p+2}}. $$