Suppose we have the following function:
$$F(x)=\sum_{n=0}^{\infty} e^{-x E_n}$$
Where $E_n$ is a positive monotonically increasing sequence, bounded from below. Is there a general condition on $E_n$ such that the function and it's analytic continuation symmetric in x? For example, consider $E_n=n+1/2$. Then:
$$F(x)=\sum_{n=0}^{\infty} e^{- x(n+1/2)}=\frac{1}{2 \sinh\big(\frac{x}{2}\big)}, x>0$$
When analytically continued to $F(-x)$, this is clearly symmetric in x and odd. Are there any other such examples?