  The behavior of the second-order Lagrangian structure functions on
state-of-the-art numerical data both in two and three dimensions is studied. On
the basis of a phenomenological connection between Eulerian space-fluctuations
and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov
$4/5$-law into a relation predicting the linear (in time) scaling for the
second order Lagrangian structure function. When such a function is directly
observed on current experimental or numerical data, it does not clearly display
a scaling regime. A parameterization of the Lagrangian structure functions
based on Batchelor model is introduced and tested on data for $3d$ turbulence,
and for $2d$ turbulence in the inverse cascade regime. Such parameterization
supports the idea, previously suggested, that both Eulerian and Lagrangian data
are consistent with a linear scaling plus finite-Reynolds number effects
affecting the small- and large-time scales. When large-time saturation effects
are properly accounted for, compensated plots show a detectable plateau already
at the available Reynolds number. Furthermore, this parameterization allows us
to make quantitative predictions on the Reynolds number value for which
Lagrangian structure functions are expected to display a scaling region.
Finally, we show that this is also sufficient to predict the anomalous
dependency of the normalized root mean squared acceleration as a function of
the Reynolds number, without fitting parameters.
