Sharpenings of Li's criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients ${\lambda }_n$ used in Li’s criterion for the Riemann Hypothesis. For $n\to \mathrm{\infty }$ we obtain that if (and only if) the Hypothesis is true, ${\lambda }_n\sim n(A\log n+B)$ (with $A>0$ and $B$ explicitly given, also for the case of more general zeta or $L$-functions); whereas in the opposite case, ${\lambda }_n$ has a non-tempered oscillatory form.