You should be able to get away with 1 relay to drive the siren. Put power to one side and the relay on the other to ground. Connect power to one side of the relay through a momentary NC switch. Ground from any detector through a diode will pull in the relay sounding the alarm. Also connect that ground through another diode so the relay locks until the cancel switch is pushed. If you connect a LED from the input side of the diode to power through the appropriate resister (about 330 to 470 ohms), you can even see active zone sources.
Since I don’t see from your drawing how much current a detector can sink, you may need to include either a transistor driver or a very low current input relay for each zone.