The disproof of Mertens' conjecture (cited above) was certainly a computations tour de force using explicit values of the zeros of $\zeta(s)$. Another good example is the paper of Rosser and Schoenfeld "Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$" Math. Comp., v. 29 1975, pp. 243-269.
We know by the Prime Number Theorem that $\Psi(x)\sim x$. Rosser and Schoenfeld use values of zeros of $\zeta(s)$ to show, for example, that for $\log(x)>105$, we have $|\Psi(x)-x|<x\epsilon(x)$, where, for $X=(\log(x)/9.6459 08801)^{1/2}$
$$
\epsilon(x)= 0.257634 \left(1 + \frac{0.96642}{x} frac{0.96642}{X} \right) x^{3/4}\exp(-x)X^{3/4}\exp(-X).
$$
The paper contains a number of results of this flavor, about the Chebyshev function $\theta(x)$, and about asymptotics of the $n$th prime $p_n$.
The reason it is difficult to convert results about low lying zeros to results about small primes is that the Explicit Formula, (mentioned in comments above) has the primes and zeros lying on opposite sides of a Fourier Transform. The Heisenberg Uncertainty Principal Principle applies
http://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle
