Euler appear at a time

For all $n\in\mathbb N$ , define the limit $L(n)=\lim_{x\to 0^+}\left(\sum_{k=1}^{n} \sqrt[x]{k}\right)^x$ Let $\alpha$ be the number of positive integral solutions for the equation $M^{\beta}=kL(n)\left(k+L(n)\right)$ where $M>1,k,\beta>1 \in\mathbb N$ and $\lim_{y\to \alpha } \sqrt[y]{y!!} =\sqrt{b\lim_{y\to\alpha} \sqrt[y]{y!}}=\frac{\sqrt{b}}{e^{\frac{\gamma}{b}}}$ where $b$ is a positive integer. Find the value of $b^{12}$ .

Notation: $\gamma$ is Euler-Mascheroni constant and $e$ is Euler's number .

This is an original and Inspired problem .