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Is it possible to sum this analytically in any way?

The sum I am looking for is the following sum as $M \to \infty$: $$ L(\omega) = \sum_{m = 1}^{M} \frac{\sin\left( N \frac{\omega_m - \omega}{2} \right)}{\sin\left( \frac{\omega_m - \omega}{2} \righ …