Because of a low mechanical loss, thin films made of silicon nitride (Si3N4) are interesting for fundamental research and development in the field of gravitational-wave detection. Si3N4 membranes allow for the characterization of quantum radiation pressure noise (RPN), which will be a limiting noise source in gravitational-wave detectors of the second and third generations. Furthermore, Si3N4 is an interesting material for possible thermal noise reduction in highly reflective mirror coatings. For both applications, the optical absorption of Si3N4 needs to be low. This paper presents absorption measurements on low-stress Si3N4 membranes showing an absorption a factor of 7 lower at 1550 nm than at 1064 nm resulting in an estimated 2 times higher sensitivity in RPN experiments at the higher wavelength and making Si3N4 an interesting material for highly reflective multimaterial mirror coatings at 1550 nm.