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.