Given an elliptic curve $E/\mathbb{Q}$, is it possible to determine whether or not $E$ has torsion points just by looking at it's Hasse-Weil L function $L(E,s)$? In general, what effects does an elliptic curve having torsion points have on its L function?

The effect of having free points is clearly seen in the Birch and Swinnerton Dyer Conjecture which relates the order of the zero at $s=1$ of $L(E,s)$ to the rank of $E$. The BSD conjecture also relates the value of $L(E,1)$ to the order of the torsion group, but looking at $L(E,1)$ is not enough to determine the order $E(\mathbb{Q})_{\mathrm{tor}}$ because of all of the other invariants involved in the expression.