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 


$$f(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)\sum\_{n=1}^{\infty}\frac{a\_{n}}{n^{s}}$$


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


$$f^{(k)}\left(1-2\prod\_{j=1}^{k}s\_{j}\right)=f^{(k)}\left(1-2\prod\_{j=1}^{k}t\_{j}\right)$$ 


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


Can I deduce that 


$$(t\_1,t\_2,\ldots,t\_{r})=(s\_1,s\_2,\ldots,s\_{r})$$


under some conditions on the derivatives of $f$?


The injectivity is not possible for this case. 



The Hasse-Weil L-function and some equations
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.


