Summary: The topic of this paper is non-self-adjoint second order differential operators with constant delay generated by \(-y''+q(x)y(x-\tau)\) where potential q is complex-valued function, \(q \in L^2[0, \pi]\). We establish properties of the spectral characteristics and research the inverse problem of recovering operators from their spectra when \(\tau\in(\frac{\pi}{2}, \pi)\). We prove that the delay and the potential is uniquely determined from two spectrum, firstly when \(y(0)=y(\pi)=0\) and secondly when \(y(0)=y'(\pi)=0\), of those operators. Also we will construct \(q\) and \(\tau\).