I have read in the history of how Sir Swinnerton-Dyer and Prof. Bryan Birch, have found this conjecture,in that I have found a line like this,
...heuristically the value of the Hasse-Weil L-function in the infinite product at $s=1$ comes to be $L(E,1)=\prod_{p}\left(\frac{N_{p}}{p}\right)^{-1}$...
but I have two questions regarding that:
1) As we know that the infinite product makes sense only when $\Re(s)>3/2$ and if we plug $s=1$ it's meaningless ,and so it doesn't make any sense, my question is that how can one fix that the Hasse-Weil $L$-function (that infinite product) so that it makes sense at $s=1$. I don't want the answer to be "it's by analytical continuation", I want the infinite product to make sense at $s=1$ ,but that product $\prod_{p}\left(\frac1{1-a_{p}p^{-s}+p^{1-2s}}\right)$ will never make sense as it is a established theorem by Hasse, can we code a similar product which makes sense at $s=1$,and if I call that new infinite product to be $L^*(E,s)$ then my condition is that that new product should not harm the conjectural properties, to be clear, that new product is in such a way that BSD holds good for $L^*(E,s)$, I mean that function order of vanishing gives the rank at $s=1$ ,and the constant and everything ,and the only difference is the convergence ,the new product is well defined at $s=1$ and it produces $L^*(E,s)=\prod_{p}\left(\frac {N_{p}}{p}\right)^{-1}$ not heuristically but concretely,if we can't code a new product how does that first statement makes sense by substituting $s=1$ in $L(E,s)$
2) I studied that the Hasse-Minkowski theorem is not going to hold good for cubic curves,if so why does one consider the reduction modulo prime for elliptic curves?
thank you