Not very many, if anything it will be a weak third party like Perot 1992, only having the possibility of winning a few states like NH, WY, AK, CO, AZ, etc. To be honest the L party is too extreme to have real political power.
So you're saying it should become a philosophy, a faction of one of the parties, like Rand Paul is doing, instead of being their own party?
Oh no, I'm not saying it shouldn't be a powerful party, if I got what I wanted I would like it to become the dominant party in American politics. I'm just being realistic in that most people aren't libertarians, and most people don't hold positions like getting rid of the income tax, or the fed, or the IRS, etc. (I do). It would likely be a influential third party at best, under our FPTP (first past the post) system. We would need proportional representation for the libertarian party to have real power.