explicit formulae for L-function

[mhill]

let be the L-function F(s)= \sum_{n=1}^{\infty} X(n) n^{-s} with a single pole at s=1

then my question is if one can define \frac{-F'(s)}{F(s)}= \sum_{n=1}^{\infty} a(n) n^{-s} ,

then taking an inverse Mellin transform we get

\sum_{n \le x} a(n) = x - \sum_{r} r^{-1} x^{r}

the question is , what are the a(n) ,

if X(n)=1 for every n then F(s) is just Riemann zeta and a(n) /\(n) Mangoldt function,

the question is , can the result be generalized for every X(n)