  This article considers the positive integers $N$ for which $\zeta_{N}(s) =
\sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on
earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for
$N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.
