Just to pick up on something mentioned in @MyNinthAccount's answer:

If you just want to determine whether or not $L(E, 1) = 0$, then there is another approach, which doesn't involve computing $L(E, 1)$ numerically to high precision and then arguing with inequalities. The point is that the ratio $$\frac{L(E, 1)}{\Omega^+_E}$$ is not only rational, but is computable as a rational number (using modular symbols). So there is an algorithm which will determine, rigorously and in finitely many steps, whether the analytic rank is 0 or not. The standard number theory software packages (Sage, Magma, maybe also Pari) have built-in functionality to do this.

If you find by this method that $L(E, 1) = 0$, then determining what the analytic rank actually is is a much harder problem.

Just to pick up on something mentioned in @MyNinthAccount's answer:

If you just want to determine whether or not $L(E, 1) = 0$, then there is another approach, which doesn't involve computing $L(E, 1)$ numerically to high precision and then arguing with inequalities. The point is that the ratio $$\frac{L(E, 1)}{\Omega^+_E}$$ is not only rational, but is exactly computable as a rational number (using modular symbols). So there is an algorithm which will determine, rigorously and in finitely many steps, whether the analytic rank is 0 or not. The standard number theory software packages (Sage, Magma, maybe also Pari) have built-in functionality to do this.

If you find by this method that $L(E, 1) = 0$, then determining what the analytic rank actually is is a much harder problem.