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.



What effect do Torsion points have on an Elliptic Curve's L function?
This has been alluded to in one of the comments, but if $E(\mathbb Q)$ has an $\ell$-torsion point, then at every prime $p$ of good reduction we have
$$ p+1-a\_p = \#E(\mathbb F\_p) \equiv 0 \pmod \ell, $$
so the local factor of the $L$-function at $p$ satisfies
$$ L\_p(T) = 1 - a\_p T + p T^2 \equiv 1 - (p+1) T + p T^2= (1-T)(1-pT) \pmod\ell. $$
Of course, for $L(E/\mathbb Q,s)$, we need to evaluate this at $T=p^{-s}$, and then the meaning of this congruence gets a bit dicey, especially when we multiply them over all $p$ to get $L(E/\mathbb Q,s)$. On the other hand, if you just want to think about the reduction curve $\tilde E\_p/\mathbb F\_p$, then its zeta function is
$$ Z(\tilde E\_p/\mathbb F\_p) = \frac{1-a\_pT+pT^2}{(1-T)(1-pT)} \equiv 1 \pmod\ell, $$
i.e., the $\ell$-torsion point means that the local zeta function is congruent to 1 modulo $\ell$.


