I am familiar with the following integral for a quotient of squares of sines in terms of the Bessel function of the first kind $J$: $$ \int_{0}^{\pi}\frac{\sin^{2}\left(\frac{N}{2}\cos(\theta)\right)}{\sin^{2}\left(\frac{1}{2}\cos(\theta)\right)}\,d\theta=\pi\left(N+2\sum_{n=1}^{N-1}(N-n)J_{0}(n)\right) $$ However, this is clearly only valid when $N$ is a positive integer. I am keen to work out the value of $$ \int_{0}^{\pi}\frac{\sin^{2}\left(\frac{L}{2}\cos(\theta)\right)}{\sin^{2}\left(\frac{1}{2}\cos(\theta)\right)}\,d\theta $$ where $L$ can be any positive number. For context, the integral comes up in calculating the diffraction patterns of periodic structures.
Thanks in advance for any help.