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I once heard a conjecture that a primitive L-function does not have multiple zeros except the central point of the critical strip.

Question:Why it is reasonable to conjecture a primitive L-function does not have multiple zeros except the central point of the critical strip? Is it possible to explain why the central values behave so different?

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About why the central point is special: first, from an elementary viewpoint, if the functional equation has sign $+1$ and the $L$-function vanishes at all, then obviously it vanishes doubly, etc. Second, much more seriously, for example the Birch-SwinnertonDyer conjecture predicts that the order of vanishing of an elliptic curve (over $\mathbb Q$) 's $L$-function at the central point is the rank of the free part of rational points of the elliptic curve. Examples are already known where this rank is $>1$. –  paul garrett Dec 23 '11 at 18:59
    
That is to say,the multiple zeros carry some arithmetical information we do not fully understand(for a higher degree L-function)? –  zy_ Dec 24 '11 at 5:21

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