A set of positive integers $d_1, \dots, d_n$ describe a n-dimensional tetrahedron $T$ with the vertices $$ (0,\dots,0), (1/d_1,0,\dots,0), (0, 1/d_2,\dots,0), \dots, (0,\dots,1/d_n).$$ Let $L_T(t)$ be the Ehrhart (quasi-)polynomial of $T$, i.e. the number of integer lattice points in $tT$.

I understand that typically everything beyond the first, second and last coefficients of $L_T(t)$ are rather difficult to determine in general. Is there an approximation possible for the other coefficients? As this for a practical problem, I'm not so much interested in an upper and lower bound, but rather a good "guess".