  It is commonly believed that the normalized gaps between consecutive
ordinates $t_n$ of the zeros of the Riemann zeta function on the critical line
can be arbitrarily large. In particular, drawing on analogies with random
matrix theory, it has been conjectured that $$\lambda' ={ lim ~ sup} ~( t_{n+1}
- t_n ) \frac{ \log( t_n /2 \pi e)}{2\pi}$$ equals $\infty$. In this article we
provide arguments, although not a rigorous proof, that $\lambda'$ is finite.
Conditional on the Riemann Hypothesis, we show that if there are no changes of
branch between consecutive zeros then $\lambda' \leq 3$, otherwise $\lambda'$
is allowed to be greater than $3$.
  Additional arguments lead us to propose $\lambda'\leq 5$.
  This proposal is consistent with numerous calculations that place lower
bounds on $\lambda'$. We present the generalization of this result to all
Dirichlet $L$-functions and those based on cusp forms.
