  In this article, we introduce a recurrence formula which only involves two
adjacent values of the Riemann zeta function at integer arguments. Based on the
formula, an algorithm to evaluate $\zeta$-values(i.e. the values of Riemann
zeta function) at odd-integers from the two nearest $\zeta$-values at
even-integers is posed and proved. The behavior of the error bound is
$O(10^{-n})$ approximately where $n$ is the argument. Our method is especially
powerful for the calculation of Riemann zeta function at large argument, while
for smaller ones it can also reach spectacular accuracies such as more than ten
decimal places.
