Asymptotic analysis of the finite product

I have the following product$$\prod _{n=1}^{\frac{L}{2}} \frac{e^{-\frac{2 \cos \left(\frac{\pi n}{L+1}\right)}{T}}+1}{e^{-\frac{2 \cos \left(\frac{\pi n}{L}\right)}{T}}+1}$$

I am interested in understanding the $L\to \infty$ limit first and then the product's value as a function of parameter $T$. I can see that the double limit $L\to \infty$ and then $T\to \infty$ the product is 1. But I can only numerically see that in the limit $L\to \infty $ and then $T\to 0$ gives me $\frac{1}{\sqrt 2}$ but if the limit is exchanged i.e. $T\to 0$ first and then $L\to \infty$ one gets $\frac{1}{2}$.

Is there a way to analyze this product in the limit $L\to \infty$ to get the full function of T or at least the correct asymptotic values at $T\to 0$ and to show that the order of the limit matters?