That paper doesn’t say how many mutations it takes to “evolve the same function”(twice?).
Rather, that paper is usually being bandied about to show that the frequency of amino acid sequences 150 residues (or more) in length capable of adopting functional enzyme folds, is as low as 1 in 10^77. Not really the same thing.
And in any case, the ludicrous conclusions typically drawn from that paper by ID-creationists (extending the work to apply to basically all proteins of a similar size) have been comprehensively debunked numerous times.
Interestingly, Discovery Institute fellow Ann Gauger has even somewhat walked back the specific claim one is bound to hear from cdesign proponentsists concerning that paper, on this very forum. She wrote:
Ann Gauger: Doug’s paper showed the rarity of a functional protein with a particular activity (B-lactam) and a particular structure ( TEM-1 B-lactam) (that’s what he and I mean by a functional fold BTW). Out all possible protein structures only 1 in 10^77 will have that structure and that enzymatic activity. It’s a way of answering the question, how many ways are there to make a protein that has that particular structure with that particular chemistry out of all possible proteins.
You will notice(well, if you can think) that this tells us nothing about how difficult it is to evolve the function. Nor even the fold, as there could be many selectable paths from shorter peptides with the same, or different functions, to this one. The sequences that adopt this fold and perform this function could also overlap sequences performing other functions in sequence space, and hence some particular rare fold+function combination might be reached easily by selection from another function that itself is much easier to evolve. Simply put, none of these possibilities have been ruled out, and hence none of the conclusions you will hear in the IDcreationist literature based on Axe’s 2004 paper can be supported. His experiment is simply incapable of supporting ANY of those conclusions. Creationists simply assume this as some sort of axiom.
And this is all without even going into @Art’s excellent 2007 article that explains why Axe’s experiment is incapable of supporting the number he extrapolates from his limited data.