Sharpenings of Li's Criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients λn used in Li's criterion for the Riemann Hypothesis. For n → ∞ we obtain that if (and only if) the Hypothesis is true, λn ∼ 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, λn has a non-tempered oscillatory form