  We prove that the dimension of the $\mathbb{Q}$-linear span of
$1,\zeta(3),\zeta(5),\ldots,\zeta(s-1)$ is at least $1.009 \cdot \log s/(1+\log
2)$ for any sufficiently large even integer $s$ and we claim that the constant
$1.009$ can be improved further to $1.108$. This slightly refines a well-known
result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof
only involves inserting the arithmetic observation of Zudilin (2001) into the
original proof of Ball-Rivoal. Although our result is covered by a recent
development of Fischler (2021+), our proof has the advantages of being simple
and providing explicit non-vanishing small linear forms in $1$ and odd zeta
values.
