A theoretical investigation of the electronic structure of the K(2) molecule, including spin-orbit effects, has been performed. Potential energies have been calculated over a large range of R up to 75a(0) for the 88 Omega(g,u)(+/-) states dissociating adiabatically into the limits up to K(4p (2)P(3/2))+K(4p (2)P(3/2)). Equilibrium distances, transition energies, harmonic frequencies, as well as depths for wells and heights for barriers are reported for all of the bound Omega(g,u)(+/-) states. Present ab initio calculations are shown to be able to reproduce quite accurately the small structures (wells and barrier) displayed at very long-range (R>50a(0)) by the (2,3)1(u) and (2)0(g)(-) purely long-range states. As the present data could help experimentalists, we make available extensive tables of energy values versus internuclear distances in our database at the web address http://www-lasim.univ-lyon1.fr/spip.php?rubrique99.