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We present a fully ab initio based scheme to compute electronic transport properties, i.e. the electrical conductivity σ and thermopower S, in the presence of electron-phonon interaction. We explicitly investigate the k-dependent structure of the Éliashberg spectral function, the coupling strength, the linewidth and the relaxation time τ. We obtain a state-dependent τ and show its necessity to reproduce the increased thermopower for temperatures below the Debye temperature, without accounting for the phonon-drag effect. Despite the detailed investigations of various k- and q-dependencies, the presented scheme can be easily applied to more complicated systems.