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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. – China-Hong Kong Nov 20 '12 at 17:26
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. 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 – François Brunault Nov 20 '12 at 17:48
Thank you Francois. – China-Hong Kong Nov 20 '12 at 18:01
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. – China-Hong Kong Nov 20 '12 at 17:42
See the book Analytic Number Theory by Iwaniec and Kowalski – Stopple Nov 20 '12 at 18:29

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: See section 3.4, example 5 which deals with elliptic curves.

Another reasonable reference is Akiyama and Tanigawa:

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