The highest lowest zero of general L-functions

Journal article, 2015

Stephen D. Miller showed that, assuming the Generalized Rie-mann Hypothesis, every entire L-function of real archimedeantype has a zero in the interval12+itwith −t0<t <t0, where t0≈14.13 corresponds to the first zero of the Riemann zeta function. We give a numerical example of a self-dual degree-4 L-function whose first positive imaginary zero is at t1≈14.496. In particular, Miller’s result does not hold for general L-functions. We show that all L-functions satisfying some additional (conjecturally true) conditions have a zero in the interval (−t2, t2)with t2≈22.661.