We investigate the minimization of the energy per point E among-dimensional Bravais lattices, depending on the choice of pairwise potential to a radially symmetric function f(|x|²). We formulate criteria for and non-minimality of some lattices for E at fixed scale based the sign of the inverse Laplace transform of f when f is a superposition exponentials, beyond the class of completely monotone functions. We also a family of non-completely monotone functions having the triangular as the unique minimizer of E at any scale. For Lennard-Jones type, we reduce the minimization problem among all Bravais lattices to a over the smaller space of unit-density lattices and we establish a to the maximum kissing problem. New numerical evidence for the optimality particular lattices for all the exponents are also given. We finally design-well potentials f such that the square lattice has lower energy E the triangular one. Many open questions are also presented.