Many thanks for your time and help, Ttokoro, and Werner. And I would like to have also the intersection-points on the green vertical lines :
Why? The vertical lines are NOT part of the cotan graph! They are graphics errors and go away if you chose "draw" instead of "line" (see the sheet attached in my previous post).
These vertical lines are simply the asymptotes at multiples of pi. So if you really want to plot these points as well, simply evaluate ln(x) at every integer multiple of pi.