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Résumé - AbstractLet K=ℚ(-D) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c>0 such that for sufficiently large D at least c·h∏ p∣D (1-p -1 ) of the h distinct L-functions L K (s,χ) do not vanish at the central point s=1/2.

We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at . We give two separate proofs of this result, with the more general method not restricted to . We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence.