  The purpose of this paper is to determine the main properties of Laplace
contour integrals $$\Lambda(z)=\frac1{2\pi i}\int_\CC\phi_L(t)e^{-zt}\,dt,$$
that solve linear differential equations
$$L[w](z):=w^{(n)}+\sum_{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0.$$ This concerns, in
particular, the order of growth, asymptotic expansions, the
Phragm\'en-Lindel\"of indicator, the distribution of zeros, the existence of
sub-normal and polynomial solutions, and the corresponding Nevanlinna
functions.
