my question is that 


already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many rational points on it,


so my question is that imagine an elliptic curve $E/\mathbb{Q}$ which has rank $r>0$ and with $ \left|{E(\mathbb{Q})}\right|=\infty$ so we find that $L(E/\mathbb{Q},s)\_{s=1}=0$


but i am interested in the zeroes on the line $s=1+it$, we know that above curve has got a zero at $s=1$
so are there any zeroes on the line $s=1+it$ ,if so tell me the cardinality of set of zeroes ,i mean whether there are finite zeroes or infinitely many zeroes 


and if i get the answer there is a deep intuition behind the answer and properties of elliptic curves,


and may be someone can conjecture still more things knowing the zeroes there on the line 


thank you,
touch everyone's feet who helped me,by suggesting books,and resources and making me what i am today by studying privately



on the Zeroes of Hasse -weil L-function 
The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T 
\ }$ zeros in the strip with $0 < t < T$. See section 5.3 of Iwaniec and Kowalski's "Analytic Number Theory," in particular Theorem 5.8.


It should be possible, if it hasn't been done already, to show that a positive proportion of these zeros are on the critical line using Selberg's method. Hafner extended Selberg's method to various families of degree 2 $L$-functions in a series of papers in the 1980s.


