Following our work on the graph of the Weierstrass function, in the spirit of those of J. Kigami and R. S. Strichartz, which enabled us to build a Laplacian on the aforementioned graph, it was natural to go further and give the related explicit resistance metric. The aim of this work is twofold. We had a special interest in the study of the spectral properties of the Laplacian. In our previous work, we have given the explicit the spectrum on the graph of the Weierstrass function. In the case of Laplacians on post-critically finite fractals, results of J.Kigami and M. Lapidus, or R. S. Strichartz, make the link between resistance metric, and asymptotic properties of the spectrum of the Laplacian, by means of an analoguous of Weyl's formula. So we asked ourselves wether those results were still valid, for the graph of the Weierstrass function.