Hello everyone,

After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes the location of finding zeroes of a $L$-function with $\chi$ character.

So in particular Swinnerton-Dyer's conjecture states that

The underlying Mordell-Weil group of an elliptic curve has an infinite cardinality if there is a zero at $s=1$. i.e. $$E(\mathbb{Q})=\infty \iff L(E,1)=0.$$ $$E(\mathbb{Q})<\infty \iff L(E,1)\neq 1.$$ So can we predict that Hasse-Weil L-function of an elliptic-cruve satisfies the GRH. After normalizing can we put it this way :

There are infinitely many zeroes in the critical strip of Hasse-Weil L-function $L(E,s). $ i.e. Let $\mathfrak{K} $ be the number of zeroes of the Hasse-Weil L-function. Then $\mathfrak{K}=\infty \iff s=1+it$ ? . ( Assume that $E$ has infinitely many points, other wise $L(E,s)\neq0$.

I have some more set of questions concerning the significance of zeroes . They can be stated as

- Are there any zeroes existing in the critical strip of $L(E,1+it)$ ? .
- We know that $\rm{Rank(E(\mathbb{Q}))}= \rm{ord}_{s=1} L(E,s).$ So what about the significance of order of vanishing for other zeroes which are located at $s=1+it$ . Do they have some interesting relation with the properties of elliptic curves ? .

Are there any interesting results that are published in this direction so far ?

Thank you.