# What is the current state of studying values of L-Functions at positive integers?

For example, the study of the enigmatic $\zeta\left(2n+1\right)$ (or the mysterious values for all primitive Dirichlet Character L-Functions $L\left(\chi, 2n+\frac{\chi\left(-1\right)+1}{2}\right)$. How far have mathematicians made it since Euler in terms of these values?

• This thread talks about $\zeta(2n+1)$ some. – Mark Jan 6 '17 at 2:08
• A lot of the material on that thread is something that I'm pretty acquainted with, except for the one about periods and motives. Im not too sure what the "underlying geometry" of the odd zeta values even means. And I'm also interested in arbitrary automorphic L-Functions, not just the degree 1 cases. – Dave huff Jan 6 '17 at 2:18
• As usual, I'd say for Dirichlet L-functions it works the same as for $\zeta(s)$. That is to say, knowing the derivatives of $\Gamma(s)$ at $s=1/2$ is as hard as knowing them at $s \in (0,1) \cap \mathbb{Q}$. For $\zeta(s)$, the values at the negative integers are known from the Laurent expansion of $\frac{1}{e^x-1}$ in term of the Bernoulli numbers. If $f(\tau)$ is a modular form, it means knowing we known the values of $L(s,f)$ at the negative integers from the Laurent expansion of $f$ at $\tau = 0$, more generally at the cusps. – reuns Jan 6 '17 at 9:54
• Perhaps this thread (second part) could be more interesting to you math.stackexchange.com/a/1771242/300700 – nguyen quang do Jan 6 '17 at 16:18
• Thank you. Its rather frustrating not being on the level to understand the material, but It sounds cool and worth looking into more. – Dave huff Jan 16 '17 at 11:59