No. Even then they're not notable. For example there are 10^97.6 primes with 100 digits, of which not all have been found (not enough storage space in the universe), but it's trivially easy to generate one at random.
Perhaps I shouldn't have said "large" or "notable". The point is that when a prime number is discovered meeting certain criteria it is included in public databases. This makes the case of an "illegal" such prime interesting.
