  Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis,
every entire $L$-function of real archimedian type has a zero in the interval
$\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ 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 $t_1\approx
14.496$. 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 $(-t_2,t_2)$ with
$t_2\approx 22.661$.
