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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.

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up vote 6 down vote accepted

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.

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This is the right value of $L(5,-4)$. However, I was looking for a general formula for $L(5,D)$ with $D<0$ a fundamental discriminant. – emiliocba Jan 23 '12 at 20:37
I note that you asked this question a few weeks ago at MathOverflow,…. I don't have anything to add to the answers you got there, other than what you might be able to track down from references in Jolley. – Gerry Myerson Jan 23 '12 at 23:34
Don't worry. It's really curios to me that there are not places with explicit computation of this kind of numbers. My other question was more general, I was interested in one specific case. Thanks again. – emiliocba Jan 24 '12 at 0:26

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