Let $$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse-Weil $L$-function of an elliptic curve $C$ over $\mathbb{Q}$. My question is: Prove that the only negative real zeroes are at the negative integers.
Hint: This is a direct consequences of the fact that the poles of the gamma-factors in the completed $L$-function imply the existence of the zeros of $L$.