This paper is devoted to a Sturm-Liouville boundary value problem with two constant delays $\tau_1, \tau_2\in[\pi/3, \pi)$ and two potentials $ q_1, q_2\in L_2[0,\pi]$. We derive an asymptotic expansion of the characteristic function and obtain asymptotic formulas for the eigenvalues. Using the Lidskii-Sadovnichii contour integration method, we establish an explicit formula for the first regularized trace of the corresponding operator.