For the Grand Riemann Hypothesis (RH for zeros of all automorphic $L$-functions), see the (somewhat technical) answer to
I think the Generalized Riemann Hypothesis (RH for zeros of Dirichlet $L$ functions) has the most significant number theoretic consequences. In addition to those listed at
such as easy primality testing and good bounds on primes in arithmetic progressions, one also gets good lower bounds on class numbers for positive definite binary quadratic forms of discriminant $D$ (or equivalently, rings of integers in complex quadratic fields): for every $\epsilon>0$ there exists an effective constant $C(\epsilon)$ such that the class number $h(d)>C(\epsilon)|D|^{1/2-\epsilon}$.