The direct spreading measures of the Laguerre polynomials, which quantify the
distribution of its Rakhmanov probability density along the positive real line
in various complementary and qualitatively different ways, are investigated.
These measures include the familiar root-mean-square or standard deviation and
the information-theoretic lengths of Fisher, Renyi and Shannon types. The
Fisher length is explicitly given. The Renyi length of order q (such that 2q is
a natural number) is also found in terms of the polynomials parameters by means
of two error-free computing approaches; one makes use of the Lauricella
functions, which is based on the Srivastava-Niukkanen linearization relation of
Laguerre polynomials, and another one which utilizes the multivariate Bell
polynomials of Combinatorics. The Shannon length cannot be exactly calculated
because of its logarithmic-functional form, but its asymptotics is provided and
sharp bounds are obtained by use of an information-theoretic optimization
procedure. Finally, all these spreading measures are mutually compared and
computationally analyzed; in particular, it is found that the apparent
quasi-linear relation between the Shannon length and the standard deviation
becomes rigorously linear only asymptotically (i.e. for n>>1).