<p>This study proposes a novel formulation of a three-dimensional spectral model for the primitive equations, where the spectral method is applied in both the horizontal and vertical directions. We utilize scaled Laguerre functions as the vertical basis and introduce a scaling parameter that enables flexible control over the distribution of vertical grid points. We demonstrate that the optimal setting of this parameter allows the model top to be placed at significantly higher altitudes while maintaining adequate grid spacing in the upper atmosphere, thereby addressing a practical limitation of previous three-dimensional spectral models. The proposed formulation is implemented as a numerical model and validated through several standard atmospheric benchmark experiments. Comparative experiments reveal that the numerical error of the three-dimensional spectral models converges significantly faster than that of a conventional vertical finite-difference model, exhibiting the rapid error reduction characteristic of spectral methods when the vertical degrees of freedom are increased. It is also demonstrated that the computational speed of the three-dimensional spectral model becomes comparable to that of the finite-difference model, provided that the vertical spectral transform subroutines are appropriately optimized. Furthermore, we investigate the properties of gravity wave propagation within the framework of the proposed discretization, using linearized two-dimensional and nonlinear three-dimensional forms of the primitive equations. This confirms that the proposed discretization method can represent upward-propagating waves more accurately than the previous three-dimensional spectral model and the conventional vertical finite-difference model.</p>