(Sorry I'm outsider in this field.)

I need to count the number of integral points in a convex polytope in $\mathbf{R}^3$. The cones in the dual fan are not necessarily regular (does it create any problem?)

So, I need the $c_1$ coefficient of the Ehrhart polynomial. There're some formulas (complicated enough for me) but I heard one can express $c_1$ as the sum (over all the edges of the polytope) of the integral lengths of the edges times some correction factors.

1. Can someone give the formula? (In the simple English, please.) Or a reference to something very down-to-earth?
2. In fact I do not need the precise expression but only a very good lower bound. Does something like this exist?