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Let $f$ be an analytic function verfifying

$f(s)=\epsilon f(2-s)$

where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is


where $N$ is an integer and $\Gamma(s)$ is the gamma function.

Let $r$ be an integer. I have a set of equations of the form


for all $k=1,...,r$. Here $f^{(k)}$ is the $k$-th derivative of $f$.

Can I deduce that


under some conditions on the derivatives of $f$?

The injectivity is not possible for this case.

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Sorry, I misread your equation as being a product:( – Marc Palm Apr 2 '13 at 9:07
I have now given an answer. In the case that the expression is in the argument, it will be clearly no! – Marc Palm Apr 2 '13 at 9:14
Yes. You make things clearer. – China-Hong Kong Apr 2 '13 at 9:17
up vote 1 down vote accepted

No, that is impossible. The $k$-th derivative of a L function has necessarily infinitely many zeros. So you can choose $s_j$ and $t_j$ inductively such that the products give distinct zeros of $f^j$. Moreover, if one of $s_j$ is zero you can't say anything clever either, but I assume that you simply have forgotten that condition in your question.

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"Coincidentally",… – Yemon Choi Apr 5 '13 at 10:03

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