A sharpening 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. In particular, we argue that if (and only if) the Hypothesis is true, ${\lambda }_n\sim n(A\log n+B)$ for $n \to \infty$(withexplicit$A>0$and$B$). The approach also holds for more general zeta or $L$-functions.