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
$$ \prod \Big( \frac{p^2}{p^2-1} \Big)^{\frac{p}{2p+2}}. $$