  Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or
Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates
of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for
the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of
Hall, use it to show that there are infinitely many gaps between consecutive
zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 =
1.732...$ times the average spacing. Using general pair correlation results due
to Murty and Perelli in conjunction with a technique of Montgomery, we also
prove the existence of small gaps between zeros of any primitive $L$-function
of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp
form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps
between consecutive zeros of $L(s,f)$ along the critical line that are at most
$< 0.823$ times the average spacing.
