  The paper deals with the Dirac operator generated on the finite interval
$[0,\pi]$ by the differential expression $-B\mathbf{y}'+Q(x)\mathbf{y}$, where
$$ B=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad
Q(x)=\begin{pmatrix}q_1(x)&q_2(x)\\q_3(x)&q_4(x)\end{pmatrix}, $$ and the
entries $q_j(x)$ belong to~$L_p[0,\pi]$ for some $p\geqslant 1$. The classes of
regular and strongly regular operators of this form are defined, depending on
the boundary conditions. The asymptotic formulas for the eigenvalues and
eigenfunctions of such operators are obtained with remainders depending on~$p$.
It it is proved that the system of eigen and associated functions of a regular
operator forms a Riesz basis with parentheses in the space~$(L_2[0,\pi])^2$ and
the usual Riesz basis, provided that the operator is strongly regular.
