We present exact explicit expressions for the diagonal spin-spin correlation functions $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ in the d=2 Ising model, in terms of elliptic integrals, for n up to 6. We also give a general structural formula for $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ and discuss its properties.