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 $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-function of a holomorphic cusp form for a congruence subgroup and it is entire function and have a holomorphic continuation. Also there is a **rapidly-converging series $f(s)$ expression $L(C,s)$ for any complex number $s$** given in <http://modular.math.washington.edu/books/bsd/> on page 9. 


I am interested on the real points $a∈ℝ$ such that the equation $$f^{(k)}(s)=a$$ have a finite number of real solutions $s$ for some $k$. Unfortunately, I have no idea to start.



Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$
I think that for any real number $a$ and any $k \geq 0$ there should be infinitely many solutions to $f^{(k)}(s) = a$ (where $f(s) = L(C,s)$). The idea is that (as you observed in your [previous question](https://mathoverflow.net/questions/141972/does-the-property-p-holds-true-for-the-derivatives-of-l)), we have $f(m) = 0$ for any integer $m \leq 0$ and that $f(m+1/2)$ tends to infinity (in absolute value) and alternates sign. (This should be easy to justify using that $f(s) \to 1$ as $s \to \infty$ and the functional equation.)


Now, it suffices to prove the following lemma:


Suppose that $g(x)$ is an infinitely differentiable function and $a\_{1}, a\_{2}, \ldots$ is a sequence of real numbers with $\cdots < a\_{k} < a\_{k-1} < \cdots < a\_{1}$ so that 


$\bullet$ $g(a\_{i}) = 0$ for all $i$,


$\bullet$ There is a constant $C$ so that $a\_{i} - a\_{i+1} \leq C$ for all $i$, and


$\bullet$ $g(x)$ is neither bounded from above nor bounded from below on
$(-\infty,a\_{1})$.


Then the same is true of $g'(x)$. That is there is a sequence $\cdots < b\_{k} < b\_{k-1} < \cdots < b\_{1}$ so that $g'(b\_{i}) = 0$, there is a constant $D$ so that $b\_{i} - b\_{i+1} \leq D$ for all $i$, and $g'(x)$ is neither bounded from above nor bounded from below on $(-\infty,b\_{1})$.


This lemma should be easy to prove using the mean value theorem.


