We have experimentally studied the ''pure long-range'' state, O-g(-), of K-39(2) for the first time by high resolution photoassociative spectroscopy of ultracold potassium atoms. Well-resolved vibrational levels have yielded molecular constants for this state. Analytical expressions for the potential energy curves of the two pure long-range states, O-g(-) and 1(u), are obtained. All the available theoretical and experimental molecular constants of the three special long-range potentials with extrema for all the alkali metals are summarized.