A good result from Balazard et al says that the Riemann Hypothesis is equivalent to the claim that

$$ int _ {- infty} ^ { infty} frac { log | zeta (1/2 + it) |} { frac {1} {4} + t ^ 2} mathrm {d} t = 0 $$ where $ zeta $ It's the Riemann zeta function.

**Is there an analogue of this criterion for the Generalized Riemann Hypothesis?** My classmates are not optimistic that it exists, because it seems that the evidence of Balazard and others is based fundamentally on the fact that $ lim_ {s rightarrow 1} ((s-1) zeta (s)) = 1 $, which is not true for other functions in L.

**Why are we interested in this analogue?**

**If you must know**, we have been asked to & # 39; & # 39; let's review & # 39; & # 39; (the author) a certain manuscript that claims a proof of the HR according to the criteria of Balazard and others. Strangely, we can not find any serious flaws in this, and we are fantasizing about generalizing the argument.