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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 $L''(E,1)$ ?

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  • $\begingroup$ This looks like an eigenvalue for the differential operator that maps a function to its derivative at 1. $\endgroup$ – Sylvain JULIEN Mar 9 at 10:40
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    $\begingroup$ @SylvainJULIEN The operation you mention sends a function to a number. It doesn't make sense to speak of its eigenvalues. $\endgroup$ – Wojowu Mar 9 at 10:55
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    $\begingroup$ 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 $0 \leq j \leq i$. $\endgroup$ – François Brunault Mar 9 at 11:46
  • $\begingroup$ 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. $\endgroup$ – Chris Wuthrich Mar 9 at 14:05
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    $\begingroup$ mathoverflow.net/questions/105190/… is related $\endgroup$ – Chris Wuthrich Mar 10 at 0:03

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