I have been reading "Introduction to Quantum Mechanics by David J. Griffiths" and in the chapter 6- Time-Independent Perturbation theory, when we are explaining the fine structure via relativistic correction section 6.3.1) and spin-orbit coupling(6.3.2), we find that the states with $\ell=0$ are often problematic, or in other words, the general formula fails for them. For example:
(a) In the relativistic correction, we find that the general formula we derived doesn't quite work for states with $\ell=0$ because for these states $p^2$ is hermitian while $p^4$ is not, where $p$=momentum operator.
(b) And in spin orbit coupling, we find that general formula doesn't work for $\ell=0$ states as it leads to an indeterminate form.
I am aware about why this happens mathematically but I don't know what's physically there, that is if there is something special about $\ell=0$ states?