Pair-produced scalar leptoquarks were sought in proton-proton collisions at a center-of-mass energy of 13 TeV by the ATLAS experiment. A dataset corresponding to an integrated luminosity of $3.2~\mathrm{fb}^{-1}$ delivered by the LHC was used. First and second generation scalar leptoquarks were searched for in events with at least two jets and two electrons or muons, respectively. No excess over the Standard Model prediction was observed. Limits of 1100 GeV (1050 GeV) were set at 95% CL on the first (second) generation leptoquark mass, given $\beta = 1$.