Using partially twisted boundary conditions we compute the K→π semi-leptonic form factors in the range of momentum transfers in lattice QCD with N f =2+1 dynamical flavours. In this way we are able to determine f+K(0) without any interpolation in the momentum transfer, thus eliminating one source of systematic error. This study confirms our earlier phenomenological ansatz for the strange quark mass dependence of the scalar form factor. We identify and estimate potentially significant NNLO effects in the chiral expansion that guides the extrapolation of the data to the physical point. Our main result is f+K(0)=09599(34)(+31−47)(14), where the first error is statistical, the second error is due to the uncertainties in the chiral extrapolation of the lattice data and the last error is an estimate of potential discretisation effects.