My question is: Why the author of this book (http://wstein.org/books/bsd/bsd.pdf) page 8 on Sec 1.4 choose $s$ real in despite that the variable is complex in the entire chapter. I am very confused about this fact.
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4"See [Dok04] for a more sophisticated analysis of computing $L(E, s)$ and its Taylor expansion for any complex number $s$." Computing for real $s$ is simpler, apparently. – Daniel Fischer Aug 22 '13 at 16:59
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The Birch-Swinnerton-Dyer conjecture is about the value of $L(s,E)$ at $s=1$, so real valued is even more general.
Marc Palm
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Correct. While I am nitpicking, it should be the "Birch and Swinnerton-Dyer conjecture", instead of "Birch-Swinnerton-Dyer". The latter seems to imply three people, while only two people are being credited, Bryan Birch and Sir H. P. F. Swinnerton-Dyer. – Álvaro Lozano-Robledo Aug 26 '13 at 14:30