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 1 / 2 + i t {\displaystyle 1/2+it} with t {\displaystyle t} a real number variable and i {\displaystyle i} 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.
References
Further reading Borwein, Peter B. (2008), The Riemann hypothesis: a resource for the aficionado and virtuoso alike, CMS books in mathematics, vol. 27, Springer-Verlag, ISBN 978-0-387-72125-5