I would agree with @steveking that it depends on what you want to try and where you are. I did my first few cyclocross races on a track bike with 23mm slicks because it was the only bike I owned at the time. I didn’t do well at all, but I did have a very good time. If you have some dry, fast races near you I’d say throw some Panaracer Paselas (or something) on there, let out a bunch of air, and give it a rip. You probably won’t do well and you’ll be racing your bike more than your course or the competitors, but sometimes that’s fun. And if it gets you out to a race, that’s even better. The community is really what makes cyclocross (in the US, anyway) special. My only caution would be that road bikes and road bike parts are not meant to take the abuse that cyclocross dishes out. The Allez Sprint is tough, but the hanger, wheels, and derailleur not so much. But hey, it can be done!
If you’re just looking to dip your toe in, it might also be worth asking around in your local community to see if anyone would lend you a bike for a race. We do that a lot here - it gets people hooked.
TL;DR Ride your bike, have fun, drink a beer afterward