The first nontrivial zeros of the Riemann $zeta$-function are
$\approx1/2\pm 14:13472i$. 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 zeros, and vice-versa. We prove that many types of
automorphic $L$-functions have a lower first zero than $\zetas$'s (see
Theorems 1.1 and 1.2). This is done using Weil's explicit formula with
carefully chosen test functions. When this method does not immediately
show that $L$-functions of a certain type have low zeros, we then
attempt to turn the tables and show that no $L$-functions of that type
exist. Thus, the argument is a combination of proving that low zeros
exist and that certain cusp forms do not. Consequently, we are able to
prove vanishing theorems and improve upon existing bounds on the
Laplace spectrum on $L\sp 2({\rm SL}\sb n(\mathbb {Z})\backslash{\rm
SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R}))$. These in turn can
be used to show that ${\rm SL}\sb 68(\mathbb {Z})\backslash{\rm SL}\sb
68(\mathbb {R})/{\rm SO}\sb 68(\mathbb {R})$ has a discrete,
nonconstant, noncuspidal eigenvalue outside the range of the
continuous spectrum on $L\sp 2({\rm SL}\sb 68(\mathbb {R})/{\rm SO}\sb
68(\mathbb {R}))$, but that this never happens for ${\rm SL}\sb
n(\mathbb {Z})\backslash/{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb
n(\mathbb {R})$ in lower rank. Another application is to cuspidal
cohomology: we show there are no cuspidal harmonic forms on ${\rm
SL}\sb n(\mathbb {Z})\backslash{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb
n(\mathbb {R})$ for $n<27$.