Eigenvalue lower bounds for manifold with positive Ricci curvature

Question

For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My question is what can we say about $\lambda_k$, for $k\ge 2$? btw. The Weyl law tell us the asymptotic behavior of $\lambda_k$ but to all of them.