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I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an elliptic curve over rationals. A google search do not gives important answers.

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Where do you want the series to converge rapidly? If it's the whole complex plane, such a series would be a proof of analytic continuation, which is as hard as modularity. Thus, you probably want to find the associated modular form and look for a rapidly converging series for the L-function of that modular form? –  Will Sawin Nov 20 '12 at 17:22
    
for any complex number s. I remamber that there is a rapidly converging series obtained by using Lavrik method, but I can't find it. –  Shpigle Nov 20 '12 at 17:26
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Yes, there is a rapidly exponentially converging series for L(E,s) which relies on the functional equation and thus on modularity of E. This has been implemented, see e.g. arxiv.org/abs/math/0207280 If you're more interested in values of s with high imaginary part, then you might need other methods, like the double exponential method. See Pascal Molin's PhD thesis math.jussieu.fr/~molinp/files/these.pdf –  François Brunault Nov 20 '12 at 17:48
    
Thank you Francois. –  Shpigle Nov 20 '12 at 18:01
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2 Answers 2

up vote 2 down vote accepted

Possibly the answer would be the so-called "approximate functional equation" for the $L$-function. This of course takes as input the modularity of the Hasse-Weil zeta function, and gives rapidly convergent series representing it at any point. I would expect Cremona's book on algorithms for modular elliptic curves to contain a description. Software like Pari/GP implements such algorithms (see the command elllseries).

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Thank you Denis. I think that there is a rapidly converging series obtained by using Lavrik method, but I can't find it in any reference. –  Shpigle Nov 20 '12 at 17:42
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See the book Analytic Number Theory by Iwaniec and Kowalski –  Stopple Nov 20 '12 at 18:29
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Also look at my paper: Computational methods and experiments in analytic number theory, in Recent Perspectives in Random Matrix Theory and Number Theory. Available here: http://arxiv.org/pdf/math/0412181v1.pdf See section 3.4, example 5 which deals with elliptic curves.

Another reasonable reference is Akiyama and Tanigawa:http://www.jstor.org/discover/10.2307/2584959?uid=3739448&uid=2129&uid=2&uid=70&uid=3737720&uid=4&sid=21102659822167

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