# Second derivative at 1 of L function of elliptic curve

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 $$L''(E,1)$$ ?

• This looks like an eigenvalue for the differential operator that maps a function to its derivative at 1. – Sylvain JULIEN Mar 9 at 10:40
• @SylvainJULIEN The operation you mention sends a function to a number. It doesn't make sense to speak of its eigenvalues. – Wojowu Mar 9 at 10:55
• If $i$ is odd then the $i$-th derivative of the completed $L$-function $\Lambda(E,s)$ vanishes at $s=1$. Taking $i=3$ one gets a relation, but it seems it will involve the $j$-th derivatives for every $0 \leq j \leq i$. – François Brunault Mar 9 at 11:46
• The same idea will link the first two coefficients of $L$ together even for higher order vanishing. I guess you are hoping for a number field analogue of the work of Zhiwei Yun and Wei Zhang, but I guess this is outside of reach currently. – Chris Wuthrich Mar 9 at 14:05
• mathoverflow.net/questions/105190/… is related – Chris Wuthrich Mar 10 at 0:03