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I want to know about reference of formulas for $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s} $$ for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have the Dirichlet class number formula.

I would like to have some reference for $s\geq2$. Thanks.-.

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In the last two sentences, you probably mean $s$ instead of $n$. –  Robert Kucharczyk Jan 3 '12 at 17:19
yes! Thank you Robert.-. –  emiliocba Jan 3 '12 at 17:27
Also note that you have the equality $L(s,D)=\zeta_K(s)/\zeta_{\mathbb{Q}}(s)$, where $K=\mathbb{Q}(\sqrt{D})$. Hence at least for even $s$, you may equally well ask for values of the Dedekind zeta function. Siegel has shown long ago that $\zeta_K(s)$ is a rational number for negative odd $s$, which should give you strong arithmetic information on $L(s,D)$ for positive even $s$, via the functional equation. –  Robert Kucharczyk Jan 3 '12 at 17:31
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4 Answers

up vote 8 down vote accepted

Let $\chi$ be any Dirichlet character modulo $q$, and let $m$ be a positive integer. Then Theorem 4.2 of Washington's Introduction to Cyclotomic Fields states that

$$ L(1-m,\chi) = - \frac{q^{m-1}}{m} \sum_{a=1}^q \chi(a)B_{m}(\tfrac{a}{q}). $$

Here $B_m(x)$ is the usual Bernoulli polynomial, defined by

$$ \frac{t e^{Xt}}{e^t-1} = \sum_{n=0}^\infty B_n(X) \frac{t^n}{n!}.$$

As Stopple pointed out, you can use the functional equation for $L(s,\chi)$ to evaluate $L(m,\chi)$, for some values of $m$, if you know the value of $ L(1-m,\chi).$

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Also, there seem to be really interesting connections between the values of $\zeta_K(s)$ at positive integers and "higher regulators" of Bloch groups. For example, see this interesting paper:

H. Gangl, D. Zagier, Classical and elliptic polylogarithms and special values of L-series, The arithmetic and geometry of algebraic cycles, Banff, AB, 1998, NATO Science Series C, Mathematical and Physical Sciences, volume 548, pages 561--615; Kluwer Academic Publishers, Dordrecht, 2000.

URL: http://www.maths.dur.ac.uk/~dma0hg/banff.pdf

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The connection between values of the zeta function at positive integers and higher K-theory (see Timothy's answer) is beautifully explained by Manfred Kolster here.

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Thanks, that's a nice link! –  Timothy Foo Jan 4 '12 at 9:46
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By the functional equation, the problem is equivalent to evaluating at negative integers. Writing the $L$-function as a linear combination of Hurwitz zeta functions (the coefficients are merely the values of the character), it is enough to evaluate the Hurwitz zeta functions at negative integers. This is Theorem 12.13 in Apostol's 'Introduction to Analytic Number Theory.'

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Note that this works if $n$ and $\chi$ have the same parity (i.e. $n$ is even if $\chi(-1)=1$ and $n$ is odd if $\chi(-1)=-1$.) Otherwise $L(1-n,\chi)$ vanishes while the corresponding $\Gamma$ factor in the functional equation has a pole. –  user18237 Jan 3 '12 at 19:18
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