I'm not a lawyer, so take what I say with a grain of salt, but you're likely going to have to purchase licensing rights, because the song writing, and lyrics are what's copyrighted, not just the recording. The Beatles management often copyright claims/strikes people simply playing cover versions of the songs because of the cord progressions of the music are copyrighted.
Lyrics are copyrighted, so even if your recorded version is original, you're using an artist's original lyrics and words, so you have to license them.
Recording and performing cover songs are a great way to build your fanbase, but it’s important to do this legally to protect yourself from litigation!
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last I heard (mike shinoda purchased licensing rights to play music and cover music in his streams) the cost of said licences are anywhere in rehlm of $5000, $15,000 or $30,000ish. I'm not sure of the exact figure and I'm sure it varies, but I know to legally cover yourself, you do need licensing rights. Again, I'm not a lawyer, this isn't legal advice, this is more "do some research, look into it some more" and just sharing things I've seen and read out of general curiosity. I've got a few musician friends so I find copyright interesting.
Here's a video I found covering the topic a little more
View: https://www.youtube.com/watch?v=5s-x-pnI0bc