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What form does the Riemann hypothesis have for a global L-function?


marked as duplicate by Bruno Joyal, Sujaan Kunalan, Dominic Michaelis, Dennis Gulko, Davide Giraudo Nov 22 '13 at 9:58

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  • $\begingroup$ Here is a thread you might want to see... $\endgroup$ – J. M. is a poor mathematician May 2 '12 at 19:08
  • $\begingroup$ @J.M., that's very helpful. Thanks a lot. $\endgroup$ – global May 2 '12 at 19:21

I'm thinking of the Hasse-Weil zeta function associated with a certain elliptic curve, which has CM. I believe that if the Riemann hypothesis is valid then the Riemann hypothesis for such a Hasse-Weil zeta function is valid. Thus, I think that this problem is as difficult as Riemann hypothesis.

I want to complement my statement. Suppose that the elliptic curve E has CM.

It is known that the L-series of E is the Hecke L-series. Thus, we can show that L(E, s) is a meromorphic function over the entire complex plane. Considering Artin's conjecture, it seems that L(E, s) has no poles.

Since L(E, s) is a meromorphic function, 1/L(E, s) is also a meromorphic function. The function 1/L(E, s) can't be zero because L(E, s) has no poles.

The Hasse-Weil zeta function is given as Z(s)Z(s-1)/ L(E, s), where Z(s) is the Riemann zeta function. Therefore, if the Hasse-Weil zeta function has zeros then the all zeros are the zeros of Z(s)Z(s-1).

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    $\begingroup$ you can edit your other post and add that info to it... $\endgroup$ – draks ... Nov 22 '13 at 6:39
  • $\begingroup$ This should have been an edit to your previous answer. Now, it would be best to edit one into the other. $\endgroup$ – robjohn Sep 9 '14 at 9:17

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