The rare decays of a kaon into a pion and a charged lepton/antilepton pair proceed via a flavour changing neutral current and therefore may only be induced beyond tree level in the Standard Model. This natural suppression makes these decays sensitive to the effects of potential New Physics. The CP conserving K→πℓ^+ℓ^- decay channels however are dominated by a single photon exchange; this involves a sizeable long-distance hadronic contribution which represents the current major source of

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... cal uncertainty. Here we outline our methodology for the computation of the long-distance contributions to these rare decay amplitudes using lattice QCD and present the numerical results of the first exploratory studies of these decays in which all but the disconnected diagrams are evaluated. The domain wall fermion ensembles of the RBC and UKQCD collaborations are used, with a pion mass of M_π∼ 430 MeV and a kaon mass of M_K∼ 625 MeV. In particular we determine the form factor, V(z), of the K^+→π^+ℓ^+ℓ^- decay from the lattice at small values of z=q^2/M_K^2, obtaining V(z)=1.37(36), 0.68(39), 0.96(64) for the three values of z=-0.5594(12), -1.0530(34), -1.4653(82) respectively.