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

Possibly the answer would be the socalled "approximate functional equation" for the $L$function. This of course takes as input the modularity of the HasseWeil 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). 


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 

