Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Question

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry on the imaginary $x$ axis. Does there, in general, exist a closed form expression for the analytic continutation of $f(x)$ to $f(-x)$?