Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$? Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$.
$$ 
\begin{array}{c|ccr}
& 0 & 1 & 2 \\ \hline
\chi_1 & 0 & 1 & 1 \\
\chi_2  & 0 & 1 & -1
\end{array}
$$
I read the L-functions for these series have special values


*

*$ L(2,\chi_1) \in \pi^2 \sqrt{3}\;\mathbb{Q} $

*$ L(1,\chi_2) \in \pi \sqrt{3}\;\mathbb{Q} $


In other words, these numbers are $\pi^k \times \sqrt{3} \times \text{(rational number)}$.  Is there a way to derive this similar to to the famous $\zeta(2) = \tfrac{\pi^2}{6}$ formula?

Here are 14 proofs of $\zeta(2) = \tfrac{\pi^2}{6}$ for reference

After thinking about it for a while:
$$ L(\chi_1,2) = \tfrac{1}{1^2} + \tfrac{1}{2^2} + 0 +  \tfrac{1}{4^2} + \tfrac{1}{5^2} + 0 +  \dots = \sum \frac{1}{n^2} - \sum \frac{1}{(3n)^2}
= \left( 1 - \frac{1}{9}\right) \sum \frac{1}{n^2} = \frac{8}{9} \zeta(2)$$
