  First measurements are presented of the diffractive cross section $\sigma_{ep
\rightarrow eXY}$ at centre-of-mass energies $\sqrt{s}$ of 225 and 252 GeV,
together with a precise new measurement at $\sqrt{s}$ of 319 GeV, using data
taken with the H1 detector in the years 2006 and 2007. Together with previous
H1 data at $\sqrt{s}$ of 301 GeV, the measurements are used to extract the
diffractive longitudinal structure function F_L^D in the range of photon
virtualities 4.0 <= Q^2 <= 44.0 GeV^2 and fractional proton longitudinal
momentum loss 5 10^{-4} <= x_{IP} <= 3 10^{-3}. The measured F_L^D is compared
with leading twist predictions based on diffractive parton densities extracted
in NLO QCD fits to previous measurements of diffractive Deep-Inelastic
Scattering and with a model which additionally includes a higher twist
contribution derived from a colour dipole approach. The ratio of the
diffractive cross section induced by longitudinally polarised photons to that
for transversely polarised photons is extracted and compared with the analogous
quantity for inclusive Deep-Inelastic Scattering.
