The literature of fundamental solutions is based on some fundamental functions such as Cauchy functions, Green functions, etc. Such functions and the approaches for their construction are valuable. The study aims to contribute the literature by introducing a method of such fundamental function for construction. Therefore, a linear ordinary differential equation with Caputo fractional derivative involving a coefficient in weighted Lebesgue space and locally a linear initial value condition is considered and studied by essentially using the concept of a special adjoint system of integral form. The solution of the adjoint system is the fundamental functional which enables the identification of the fundamental function for the problem.