Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that $$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$ where $\gamma$ is Euler's constant and $L(E,1)$ is given by BSD.

Can anything interesting be said about $L''(E,1)$ ?