This is a vague question: I'm sorry for that.

Let's start with $\chi$ a (primitive odd) Dirichlet character modulo $n$ and look at the corresponding L-function $$ L(s, \chi)=\sum \frac{\chi(n)}{n^s}. $$ A classical formula due to Hurwitz computes the logarithmic derivative of $L(s, \chi)$ at $s=0$ in terms of the Gamma function: $$ \frac{L'(0, \chi)}{L(0,\chi)}=-\log n + \frac{\sum \log \Gamma(1-\frac{u}{n})\chi(u)}{\sum \chi(u) \frac{u}{n}} $$ (By the way, does anybody know of some good references for this formula?)

That's the abelian case. Now imagine $L/\mathbb{Q}$ is a Galois field extension with non-abelian Galois group $G$, and let $\chi$ be a character of $G$. Now one has the Artin L-function.

**Question**: Is there any similar formula know for $\frac{L'(0, \chi)}{L(0, \chi)}$?

My naive guess would be the following: by Brauer's theorem one knows that $\chi$ can be written as a sum with rational coefficients of characters induced by cyclic subgroups of $G$, so $\log L(s, \chi)$ should be a sum with rational coefficients of some Dirichlet L-functions and for each one there is a such a formula. However, I was unable to find something like that in the literature. Is there some mistake here? If rational coefficients create some trouble, one can also write $\chi$ as a combination with integer coefficients of characters induced from elementary subgroups. Does it help?

Another question: is there any relation between Jacobi sums and these derivatives?

Thanks for your help.