The highest lowest zero of general L-functions

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Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire L-function of real archimedian type has a zero in the interval 12+i t with -t₀ < t < t₀, where t₀≈ 14.13 corresponds to the first zero of the Riemann zeta function. We give an 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.