Wood, see the article by Kumar Murty in Seminar on Fermat's Last Theorem. He shows how the L-series converges (conditionally!) on Re(s) > 5/6, thus in particular at s = 1. You can find the book on Google books and do a search on "5/6" to find the page. OK, I just did that and will tell you: it's on page 15. He proves the theorem for any Dirichlet series converging abs. for Re(s) > 3/2 and having an analytic continuation and suitable functional equation relating values at s and 2-s. (Thus in practice it is a theorem about L-functions of suitable weight 2 modular forms, certainly nothing *directly* about elliptic curves!) He also says that what he describes is a special case of a more general result, with citation.

Where the Dirichlet series converges, it still represents the L-function that may have been analytically continued to the wider region by some other method, since Dirichlet series are analytic on the half-plane to the right of any point where they converge and moreover at any point where they converge the value is the limit of the function taken along the line to the right of the point (Abel's theorem for power series on discs works for Dirichlet series on right half-planes). So we are assured that if you happen to know the series itself converges somewhere new it still equals the orthodox analytic continuation (whatever that means).

On the other hand, the Euler product has surprises. Goldfeld discovered that if the Euler product for an ell. curve over Q converges at s = 1, in the natural sense of partial Euler products over primes up to x as x goes to $\infty$, and the value of the Euler product is nonzero, then this value is *not* L(1) but rather is off from this by a factor of $\sqrt{2}$. Of course there was no real input about elliptic curves directly: Goldfeld was *assuming* the ell. curve was modular (he was writing in the 1980s) and used that right away.
It turns out exactly the same thing happens for *quadratic* Dirichlet L-functions at s = 1/2: if the partial Euler product at s = 1/2 converges to a nonzero value then again you're off by a factor of $\sqrt{2}$ from L(1/2), but in one setting the factor is $\sqrt{2}$ and in the other it's $1/\sqrt{2}$. For non-quadratic Dirichlet L-functions there's no funny business: if the Euler product converges at s = 1/2 to a nonzero value (which, by the way, it always should since nobody expects Dirichlet L-functions vanish at 1/2) then the value will be L(1/2). I first heard about Goldfeld's result at a talk by Karl Rubin (he gave the usual heuristic for BSD conjecture by looking at the Euler product at s = 1 and some wiseguy in the audience asked if it really did converge at s = 1 and Rubin mentioned there was a paper of Goldfeld on that), but when I read Goldfeld's paper I was confused by part of it, so in trying to work it out in the simpler example of Dirichlet L-functions I wound up seeing I could prove the same kind of theorem for any Euler product over a global field having the properties everyone expects it should have. This turns out to be related to properties of symmetric and exterior square Euler products and is morally the same quadratic bias (called Chebyshev's bias) that Sarnak and Rubinstein found when they worked out comparative statistics on the number of primes up to x in different congruence class mod $m$: classes of squares or non-squares exhibit different fine growth rates compared to one another. For more details on the partial Euler products, including some numerical examples, see my paper http://www.math.uconn.edu/~kconrad/articles/eulerprod.pdf.

By the way, one can definitely observe this $\sqrt{2}$ business happening numerically but we'll never expect to *prove* it happens since it actually implies the Riemann hypothesis for the relevant L-function. If the product converges to a nonzero value at a point on the critical line it is no real surprise that you could prove the Dirichlet series for the log of the Euler product converges everywhere to the right of the critical line, which implies the Euler product itself converges to a nonzero value everywhere to the right of the critical line, hence Riemann hypothesis. In fact, as I show in that paper, this (suitable) Euler product convergence at a point on the critical line is actually equivalent to something which at present appears to lies deeper than the Riemann hypothesis but is still plausible. There's no lack of results which imply RH but are themselves false, you see. This probably isn't one of them.

In summary, the Dirichlet series for ell. curve L-functions (over $\mathbf Q$) can provably be shown to converge on a wider region than they are usually said to converge and still equal the L-function there, including s = 1, while the Euler product probably does converge at s = 1 too but if the value is not 0 then it's not going to converge to what you expect... unless you think the Riemann hypothesis is false.