The Jagiellonian Positron Emission Tomograph (J-PET) is a multi-purpose device built out of plastic scintillators. With large acceptance and high angular resolution, it is suitable for the studies of various phenomena such as discrete symmetries in decay of positronium atom or entangled states of photons as well as medical imaging. J-PET enables the measurement of momenta together with photon polarization related observables. Large acceptance and high granularity of the J-PET detector enables measurement of ortho-positronium decays into three photons in the whole phase space. In this paper, we present the search of the C-forbidden decays of positronium with the J-PET detector.