@Mr. It depends on the number $n$. When you take $n$ to be prime for example, $G(n)$ can get arbitrarily small. On the other hand, if you take $N_k$ to be the least common multiple of the first $k$ positive integers, then $N_k/G(N_k)$ does converge to $e^{\gamma}$

I really don't see how it does. Those 3 theorems don't mention Linnik's Theorem and specifically talk about 'sufficiently large' moduli.

The Wikipedia article says 'Linnik's proof showed c and L to be effectively computable' and 'in Heath-Brown's result the constant c is effectively computable.' And, for example, the most recent paper by Matti Jutila that is mentioned and referenced on the Wikipedia page also states 'everything can be made explicit. In fact it would not be too difficult to calculate'. But I have not been able to actually find such a calculation yet.