In mathematics, the grand Riemann hypothesis is a generalisation of both the Riemann hypothesis and the generalized Riemann hypothesis. It states that the non-trivial zeros of all automorphic L-functions lie on the critical line with a real number variable and the imaginary unit.
The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L-functions lie on the critical line or the real line.
Notes Robert Langlands, in his general functoriality conjectures, asserts that all global L-functions should be automorphic. The Siegel zero, conjectured not to exist, is a possible real zero of a Dirichlet L-series, rather near s = 1. L-functions of Maass cusp forms can have trivial zeros which are off the real line.
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