How did Birch and Swinnerton Dyer arrive at their conjecture? I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$.  Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product  $\prod_p (\frac{1}{1-a_pp^{-s}+p^{1-2s}})$?  I think that it gives $\prod (\frac{N_p}{p})^{-1}$, which does not make sense under the usual definition of infinite product.
Note: I have posted the same at math.stackexchange but I would like further explanation beyond what was given there.
 A: For what it's worth, here are some historical comments.
Both Birch and S-D spoke in Cambridge a few weeks ago, on the history of their conjecture. To my surprise, both of them emphasized the role not of the $L$-function, but of the Tamagawa number, in their comments on how it all came about.
Tamagawa had introduced this invariant associated to a semisimple algebraic group over a number field, and one can interpret it adelically or as a product of local factors. B and S--D were trying to "compute the Tamagawa number of an elliptic curve" -- or more precisely, to see what the analogue should be in this situation. The local factors came out to be $N_p/p$, at least at the good primes. Historically, the $L$-function came later. I asked S--D why this might have been, and he said something like "Weil was pushing the $L$-function as being of central importance, so, naturally, everyone else was avoiding it like the plague". I am not sure this comment is to be taken so seriously -- this is perhaps more a reflection of S-D's sense of humour (he'd made some rather caustic comments about Mordell's bridge playing skills earlier, again probably just to get laughs, and he succeeded admirably in getting them). But you have to remember the resources available to them at the time: they were initially not thinking about the $L$-function, but they did have this access to this gigantic machine, the size of a lecture theatre, that was capable of computing the product of $N_p/p$ for all primes less than 1000, and this for them was basically a miracle, because ten years prior to that if you wanted to do this calculation then you'd better have a lot of pencils/paper handy. 
So they used what they had, they were focussed initially on Tamagawa numbers (this is I guess the reason that the fudge factors at the bad primes became known as Tamagawa factors?), they had access to a computing machine and S--D knew how to use it, and perhaps crucially one should stress that whilst we now know the $L$-function to be of central importance, it was perhaps not so clear at that time. There was no Langlands programme, there was no converse theorem -- this was the late 50s. It was only a couple of years later, when Birch was talking to Shimura, that Shimura told him that in the CM case one could actually evaluate the $L$-function at 1 in concrete cases and get concrete numbers, and then Davenport told Birch a concrete algorithm which would work and could be done by hand. I have seen with my own eyes the piece of paper in which Davenport sketched the idea to Birch, and Birch has written "keep this" on the top and underlined it! Birch then proceeds to compute various explicit examples of special values of $L$-functions on the next few pages, in the CM case, but maybe this was already after the first work had been published.
I mentioned all this to Rene Schoof yesterday and he claimed that there were pictures of the pieces of paper on youtube of all places (I know William Stein was taking pictures frantically -- my eyes were just popping out of my head -- all these really important historical documents, that Birch claimed were just gathering dust in a wardrobe at home!).
