  The first nontrivial zeroes of the Riemann $\zeta$ function are $\approx
1/2+\pm14.13472i$. We investigate the question of whether or not any other
L-function has a higher lowest zero. To do so we try to quantify the notion
that the L-function of a ``small'' automorphic representation (i.e. one with
small level and archimedean type) does not have small zeroes, and vice-versa.
We prove that many types of automorphic L-functions have a lower first zero
than $\zeta$'s. This is done using Weil's explicit formula with
carefully-chosen test functions. When this method does not immediately show
L-functions of a certain type have low zeroes, we then attempt to turn the
tables and show no L-functions of that type exist. Thus the argument is a
combination of proving low zeroes exist and that certain cusp forms do not.
Consequently we are able to prove cohomological vanishing theorems and improve
upon existing bounds on the Laplace spectrum on $L^2(\quo n)$.
