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 protonproton collision data corresponding to an integrated luminosity of 3.2 fb(-1) at root s = 13TeV 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 13TeV and 8TeV is also evaluated. Limits on the production cross section times branching ratio to two photons for the two resonance types are reported.