  For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and
$K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$
given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an
explicit expression for the $L$-functions of these curves in terms of Jacobi
sums. Moreover, we deduce from this calculation that the curves $E_d$ satisfy
an analogue of the Brauer-Siegel theorem. More precisely, we estimate the
asymptotic growth of the product of the order of the Tate-Shafarevich group of
$E_d$ (which is known to be finite) by its N\'eron-Tate regulator, in terms of
the exponential differential height of $E_d$, as $d\to\infty$.
