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2s/2p energy split, not predicted by the Dirac equation, but explained by quantum electrodynamics, which is one of the first great triumphs of that theory.
Initial experiment: Lamb-Retherford experiment.
On the return from the train from the Shelter Island Conference in New York, Hans Bethe managed to do a non-relativistic calculation of the Lamb shift. He then published as The Electromagnetic Shift of Energy Levels by Hans Bethe (1947) which is still paywalled as of 2021, fuck me: by Physical review.
The Electromagnetic Shift of Energy Levels Freeman Dyson (1948) published on Physical review is apparently a relativistic analysis of the same: also paywalled as of 2021.
TODO how do the infinities show up, and how did people solve them?
Video 6. Murray Gell-Mann - The race to calculate the relativistic Lamb shift by Web of Stories (1997) Source. Quick historical overview. Mentions that Richard Feynman and Julian Schwinger were using mass renormalization and cancellation if infinities. He says that French and Weisskopf actually managed to do the correct calculations first with a less elegant method. History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)
Video 7. Freeman Dyson - The Lamb shift by Web of Stories (1998) Source.
Mentions that he moved to the USA from the United Kingdom specifically because great experiments were being carried at Columbia University, which is where the Lamb-Retherford experiment was done, and that Isidor Isaac Rabi was the head at the time.
He then explains mass renormalization briefly: instead of calculating from scratch, you just compare the raw electron to the bound electron and take the difference. Both of those have infinities in them, but the difference between them cancels out those infinities.
Video 8. Hans Bethe - The Lamb shift (1996) Source.
Ahh, Hans is so old in that video, it is sad to see. He did live a lot tough. Mentions that the shift is of about 1000 MHz.
The following video: Hans Bethe - Calculating the Lamb shift.
Video 9. Lamb shift by Vidya-mitra (2018) Source.