Numerical theory nt. Is there an analogous to the criterion of Balazard et al for the Generalized Riemann Hypothesis?

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.