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 <math>1/2 + it</math> with <math>t</math> a real number variable and <math>i</math> 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.