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WHO you get to write your LORs means a lot less than WHAT they say. Most adcomms will be able to spot a generic or less than personal letter, especially if it comes from someone important... Unless you think that it will provide some prospective to who you are as an applicant beyond the fact that you did the internship (that's on the resume), then I would lean against doing it.

This is just my opinion, but I think two good letters from professors is enough. I doubt that a third letter from an intern coordinator with whom you only worked for a few months would help your application by much, if at all. If you want, you can request the letter and keep it on hold to submit in case you get waitlisted.

I wouldn't send it in. There's no 'prestige bump' from having the letter from the Senator's office.