  For every $p \geq 2$, we obtained an explicit construction of a family of
$\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro
algebra modules. Furthermore, we classified all irreducible self-dual
$\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and
computed their graded dimensions. In addition, we constructed certain hidden
logarithmic intertwining operators among two ordinary and one logarithmic
$\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically
precise formulation and interpretation of what physicists have been referring
to as "logarithmic conformal field theory" of central charge
$c_{p,1}=1-\frac{6(p-1)^2}{p}, p \geq 2$. Our explicit construction can be
easily applied for computations of correlation functions. Techniques from this
paper can be used to study the triplet vertex operator algebra
$\mathcal{W}(2,(2p-1)^3)$ and other logarithmic models.
