The calculation of the determinant for the second order covariant derivative operator is analyzed in the space R4 and for Yang-Mills instanton field configurations. The problems inherent to the ultraviolet ζ-function renormalization method and infrared divergencies of this operator are reviewed. A method for estimating the asymptotic coefficients of its determinant has been discussed and the failure of the general conformal invariance induced by any regularization technique has also been considered. A particular solution valid for a Yang-Mills multi instanton configuration, valuable in order to get the non conformally invariant piece of the general solution, is the main result obtained here. We give it as an integral equation, namely in a semi-explicit form.