Given a wavefunction for hydrogen $\psi(n,l,m)$ it is possible to calculate its associated energy from $E=-13.6/n^2$. Does a similar equation exist for $L^2$ and $L_z$? That is, if we are given the $n$, $l$ and $m$ values, is it possible to calculate the energy and angular momentum, without needing to know any further numbers?
The quantum numbers $l$ and $m$ only specify the degeneracies. Once the wavefunction is known, expectation value of any observable of the system can be determined. In this case,
It can be clearly seen that the energy eigenvalues are dependent only on the principal quantum number $n$ while the eigenvalues of the various angular momentum operators are dependent only on the azimuthal and the magnetic quantum numbers $l$ and $m$ respectively.