Searches for new resonances decaying into two photons in the ATLAS experiment at the CERN Large Hadron Collider are described. The analysis is based on proton-proton collision data corresponding to an integrated luminosity of 3.2 fb−1 at s=13 TeV recorded in 2015. Two searches are performed, one targeted at a spin-2 particle of mass larger than 500 GeV, using Randall-Sundrum graviton states as a benchmark model, and one optimized for a spin-0 particle of mass larger than 200 GeV. Varying both the mass and the decay width, the most significant deviation from the background-only hypothesis is observed at a diphoton invariant mass around 750 GeV with local significances of 3.8 and 3.9 standard deviations in the searches optimized for a spin-2 and spin-0 particle, respectively. The global significances are estimated to be 2.1 standard deviations for both analyses. The consistency between the data collected at 13 TeV and 8 TeV is also evaluated. Limits on the production cross section times branching ratio to two photons for the two resonance types are reported.[Figure not available: see fulltext.]