The form of your potential is
Typically we would like to set either the potential at s = 0 to be 0 or the potential at infinity to be 0. We can do neither in this case.
The whole trouble is that the line charge extends to infinity, so the constant we need to subtract to make this potential realistic is also infinite. This is not a very good Mathematical procedure, so we'd like something better.
The solution to this dilemma has exactly the same form that you wrote:
The whole point of this solution is the physical interpretation of a: The coordinate s = a is the point where we are defining the potential to be 0.
I hope this helps clear up some of the confusion.