Goldbach Division

Everyone knows the Goldbach Conjecture! Here is one version:

1) Any odd integer greater than $17$ can be written as the sum of three different odd primes. 2) Any even integer greater than $6$ can be written as the sum of two different odd primes.

Inspired by this famous mathematical problem, someone decided to study the Goldbach Conjecture in more detail and introduced a new term: Goldbach Division. If a number is even, we express it as the sum of two different odd primes; if it is odd, as the sum of three different odd primes. Such a way of expressing a given $N$ is called a Goldbach Division and is denoted as $G(N)$.

For example, if N = 18, there are two ways to express $N$ as required:

18 = 5 + 13 = 7 + 11

If N = 19, there is only one way to express $N$ as required:

19 = 3 + 5 + 11

Your task: given $N$, find $|G(N)|$, that is, the number of different Goldbach Divisions for $N$.