Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. I find the following proof in an old Russian book: I want to prove that there is no $a∈ℂ$ such that the fiber $L^{-1}(a)$ is finite. Indeed, As $s$ takes on real negative values, there are trivial zeros at the integers. The sign of $L$ changes as we pass through a zero. We can also check that $|L(-n+1/2)|$ gets large as $n$ gets large. So now use the intermediate value theorem.
This proof is completely unclear for me. Can anyone explain to me the steps of this proof.
For every $n$, let $I_n=(-n,-n+1)$. If $a\gt0$ and $L\gt0$ on some $I_n$, then $L\gt0$ on each interval $I_{n+2k}$ and the supremum of $L$ on $I_{n+2k}$ goes to infinity when $k\to+\infty$. Assume that this maximum is at least $a$ for every $k\geqslant k_a$ and let $k\geqslant k_a$. Then, $L(-n-2k)=0$ and $L(-n-2k+x_k)\geqslant a$ for some $0\lt x_k\lt1$. By the IVT, there exists $s_k$ in $(-n-2k,-n-2k+x_k]$ such that $L(s_k)=a$. Hence $L^{-1}(a)\supseteq\{s_k\mid k\geqslant k_a\}$, which is infinite.
Adapt the idea if $a\lt0$.