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We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k,l}\left(T\right)={\int }_0^T{\left|{\zeta }^{\left(l\right)}(\frac{1}{2}+it)\right|}^{2k}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k,l}\left(T\right)$.