  We prove the analogue of the Riemann-Roch formula for the noncommutative two
torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary
translation invariant complex structure and a Weyl factor represented by a
positive element $k\in C^{\infty}(\mathbb{T}_{\theta}^2)$.
  We consider a topologically trivial line bundle equipped with a general
holomorphic structure and the corresponding twisted Dolbeault Laplacians. We
define an spectral triple ($A_{\theta}, \mathcal{H}, D)$ that encodes the
twisted Dolbeault complex of $ A_{\theta}$ and whose index gives the left hand
side of the Riemann-Roch formula.
  Using Connes' pseudodifferential calculus and heat equation techniques, we
explicitly compute the $b_2$ terms of the asymptotic expansion of $\text{Tr}
(e^{-tD^2})$. We find that the curvature term on the right hand side of the
Riemann-Roch formula coincides with the scalar curvature of the noncommutative
torus recently defined and computed in \cite{CM1} and \cite{FK2}.
