Weierstrass factorization theorem for non-entire functions

Question

I know that any entire function $f(z)$ can be represented in the form of an infinite product according to the Weierstrass factorization theorem:

$$ f(z) = z^m e^{g(z)}\prod_{n=1}^\infty E_{p_n}\left(\frac{z}{a_n}\right).\quad\quad (1) $$

Suppose that I am interested in a function $f(z)$ that is holomorphic at every point of a subset $\Omega\subset \mathbb{C}$ but it is not entire.

1. Can I represent $f(z)$ as in Equation (1) in the subset $\Omega$?

2. In the negative case, is there any generalization of the above theorem to non-entire functions?