Unfamiliar Limit

Calculus Level 5

$\color{red}{\mathfrak{L}}=\displaystyle\lim_{n\to\infty}\left(\ln\left(\left(\dfrac{n^2-1^2}{n^2}\right)\cdot\left(\dfrac{n^2-2^2}{n^2}\right)\cdots\left(\dfrac{2n-1}{n^2}\right)\right)^{1/n}\right)$

If $$\color{red}{\mathfrak{L}}$$ can be expressed in the form $$\color{red}{\ln\left(\varphi \large{e}^{\alpha}\right)^{\gamma}}$$ for $$\varphi,\alpha,\gamma\in \mathbb Z-\{1\}$$, then:

$\large (\varphi+\alpha+\gamma)!=\ ?$

×