This paper aims to reveal the relationship between the minimum L 2-sensitivity of state-space digital filters and the Gramian-preserving frequency transformation. To this end, we first give a prototype low-pass state-space filter in such a manner that its structure becomes the minimum L2-sensitivity structure. Then we apply the Gramian-preserving (LP-LP) frequency transformation with a tunable parameter to this prototype filter. In this way we obtain a low-pass state-space filter with tunable cutoff frequency from a prescribed prototype low-pass filter with minimum L2-sensitivity. For this tunable low-pass state-space filter, we evaluate the L2-sensitivity over the entire range of cutoff frequencies. The evaluation result shows that, although the minimality of the L2-sensitivity is not preserved under the frequency transformation, the L2-sensitivity of the tunable filter given in this way becomes very close to the minimum value for arbitrary cutoff frequencies.