With the canonical choices of directions etc, assuming the quantum mechanical wavefunction $\Psi$ is an eigenfunction of the angular momentum operator $L_z$, we can express a rotation of angle $\alpha$ around the $z$-axis in terms of an exponential of $L_z$:
Further assuming that $\Psi$ is an eigenfunction of $L^2$, is there a rotation (potentially including all three directions) that lead to an exponential of $L^2$ (and possibly $L_z$)?
I assumed there would be a way to express a general rotation in this way, i.e. something like
where $f$ is linear in $L_z$ and $L^2$, but I haven't been able to find it.
Note that the question is not about finding an expression for the general rotation but about finding any rotation that can be expressed as an exponential that includes $L^2$ and possibly $L_z$ but no other operators or powers of these (and nothing like $0\cdot L^2$).
Also, if it exists, getting the rotation in terms of spherical coordinates $\theta$ and $\phi$ is great if that is easier.