L-function at s=5 with D=-4?

Question

I want to know the value of $L(5,-4)$. Recall that $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right) n^{-s}. $$ I would like a reference with computations of $L(5,D)$, or more generally, of $L(s,D)$ with $s$ and odd natural number.

Answer 1

If I'm not mistaken, $D=-4$ is equivalent to $D=-1$, and your symbol is just $\pm1$ depending on the resiude of $n$ modulo 4, so $$L(5,-4)=1-3^{-5}+5^{-5}-7^{-5}+\dots$$ This is known to be $5\pi^5/1536$. A reference for this and other special values of $L$-functions is the section titled Other Power Series in Jolley's book, Summation of Series, a Dover paperback.

For this particular series, Jolley cites page 501 of J Edwards, Differential Calculus, published by Macmillan in 1938.