A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that

$$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ is the zeta function of Riemann.

**Is there an analogue of this criterion for the Generalised Riemann Hypothesis ?** My classmates aren't optimistic that it exists, because it seems Balazard et al's proof relies crucially on the fact that $\lim_{s\rightarrow 1}((s-1)\zeta(s))=1$, which is not true for other L-functions.

An adelic causality problem related to abelian $L$-functions? Or, by "other $L$-functions," do you intend automorphic $L$-functions in a sweeping generality? $\endgroup$ – Vesselin Dimitrov Nov 24 '18 at 12:06