Limit of n times a sum of shifted sine values

Let $L=\lim_{n\to\infty}n\left[f\left(\frac1n\right)+f\left(\frac2n\right)+\cdots+f\left(\frac kn\right)-kf(0)\right]$, where $k$ is a positive integer. If $f(x)=\sin x$, then $L$ is equal to