  We investigate in this paper the vanishing at $s=1$ of the twisted
$L$-functions of elliptic curves $E$ defined over the rational function field
$\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and
characteristic $\geq 5$) for twists by Dirichlet characters of prime order
$\ell \geq 3$, from both a theoretical and numerical point of view. In the case
of number fields, it is predicted that such vanishing is a very rare event, and
our numerical data seems to indicate that this is also the case over function
fields for non-constant curves. For constant curves, we adapt the techniques of
Li and Donepudi--Li who proved vanishing at $s=1/2$ for infinitely many
Dirichlet $L$-functions over $\mathbb{F}_q(t)$ based on the existence of one,
and we can prove that if there is one $\chi_0$ such that $L(E, \chi_0, 1)=0$,
then there are infinitely many. Finally, we provide some examples which show
that twisted $L$-functions of constant elliptic curves over $\mathbb{F}_q(t)$
behave differently than the general ones.
