Grand Riemann hypothesis

In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-functions lie on the critical line ${\frac {1}{2}}+it$ with $t$ a real number variable and $i$ 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.

in his general functoriality conjectures, asserts that all global L-functions should be automorphic.^[1]

Robert Langlands

The , conjectured not to exist,^[2] is a possible real zero of a Dirichlet L-series, rather near s = 1.

Siegel zero

L-functions of Maass cusp forms can have trivial zeros which are off the real line.

Borwein, Peter B. (2008), The Riemann hypothesis: a resource for the aficionado and virtuoso alike, CMS books in mathematics, vol. 27, , ISBN 978-0-387-72125-5

Grand Riemann hypothesis

Robert Langlands

Siegel zero

Springer-Verlag