I have found a systematic way to find the exact value of the $L$-series $$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ for $s$ a positive even integer if $\chi(-1)=1$ and $s$ odd and positive if $\chi(-1)=-1$. Is this known? I haven't been able to find such results, other than the class number formula for $s=1$. I would appreciate if anyone would be able to help.

  • 1
    $\begingroup$ Yes, such formulas are known (in terms of generalized Bernoulli numbers). I suggest you look in Washington's Cyclotomic fields. Congratulations on your discovery! $\endgroup$ – Bruno Joyal Mar 7 '16 at 1:51
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    $\begingroup$ $\Gamma(s) L(s,\chi) = \int_0^\infty x^{s-1} \sum_{n=1}^\infty \chi(n) e^{-nx} dx$ and the problem reduces to evaluating the $k$th derivative of $x\sum_{n=1}^\infty \chi(n) e^{-nx} = \frac{x}{\textstyle e^{qx}-1}\sum_{n=1}^q \chi(n) e^{-nx}$ at $0$, using that $\frac{x}{\textstyle e^{qx}-1}$ is the generating function of the Bernouilli numbers, and the functional equaiton for $L(s,\chi)$ $\endgroup$ – reuns Mar 7 '16 at 1:57

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