The author investigates the set of odd values of the Riemann zeta-function \(\zeta(s)=\sum_{n=1}^{\infty}1/n^s\), \(s=\sigma+it\), \(\sigma>1\). It is shown that all real numbers can be strongly approximated using certain linear combinations of odd values of the function \(\zeta(s)\). More precisely: for any given integer \(n \geq 3\), there exists an integer \(r \leq n\), \(n,r \to \infty\), a sequence of rational polynomials \(s_r(x)=a_rx^r+...+a_1x+a_0\) converging in \({\mathcal C}^\infty[0,2\pi]\) under the supremum metric with coefficients depending on \(\alpha,n,q\), and satisfying the estimate \[ \bigg|\alpha+\sum_{k=1}^{[r/2]}c_k\zeta(2k+1)\bigg| \ll_{\alpha,q}\frac{1}{n^q} \] with real \(\alpha\), positive integer \(q\), and \[ c_k:=\sum_{m=2k}^{r}(-1)^k \frac{m! (2\pi)^{m-2k+1}a_m}{(m-2k+1)!} \not =0 \] for all \(1 \leq k \leq [r/2]\).