The next prime number within the same range after about $10^8$. Eratosthenes is as fast as $O(n)=n\log(\log(n))$, while the best known algorithm for Riemann zeros is far from trivial and about $O(n)=n^{1+\epsilon}$ where if you want to reduce $\epsilon$ you need more space.

The difference is that you can find a very large Riemann zeros without knowing any previous zero, while you need to find all primes before any given value if you want to be 100% sure that you have found a prime number.

You can, however, use some algorithms that are testing if a number is prime, but then you still have to scan some region in order to find a prime number. The best known deterministic is way faster than any known Riemann zeta algorithm, it is known as  AKS algorithm and has $O(\log(n)^6)$. Above that you are left with quick but only probabilistic algorithms for prime numbers.

Overall prime is a winner.