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The quadratic character experiment. (English) Zbl 1226.11089

The author develops a fast algorithm for computing the zeros of the quadratic character \(L\)-functions for all fundamental discriminants \(-d\) with \(10^{12}<d<10^{12}+10^7\). Then, he discusses the data obtained from the computation of the zeros of approximately \(3\times 10^6\) quadratic character \(L\)-functions for negative fundamental discriminants \(-d\) with \(d\) in above mentioned range. The author ends the paper by some implementation notes including error estimates and more details on his computations, as more as, he gives an Appendix on Miller’s ‘refined’ 1-level density.

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11Y16 Number-theoretic algorithms; complexity
11Y35 Analytic computations
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