Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $\text{ord}_{s=1}L(E, s)\le 1$.

What is known in the case $\text{rank} E(\mathbb{Q})\le 1$? Is it known that if $\text{rank} E(\mathbb{Q})=1$, then $L'(E, 1)\neq 0$?

Thank you!