In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character sending 1 mod 3 to 1 , 2 mod 3 to -1 and 0 mod 3 to 0.
How does one calculate this Dirichlet L-function?
Bonus question:Also, is there a way to generalise the methods in David Speyer's answer, at least for when the number alpha is a fundamental unit in a quadratic number ring that is a PID?Can someone explain why the number in the question (namely $2+\sqrt3$) has these miraculous properties (for instance, the region D becomes a fundamental one mod $\Gamma$).All this seems a bit serendipitous to me ( but then again, I'm no expert)
I have found a very easy proof from Jack D'Aurizio's answer here:
With a similar technique: $$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$ so: $$\color{red}{L(1,\chi_2)}=\int_{0}^{1/2}\frac{dx}{x^2+3/4}=\frac{1}{\sqrt{3}}\arctan\sqrt{3}=\color{red}{\frac{\pi}{3\sqrt{3}}.}$$