This paper presents a unified framework based on hierarchical differential algebraic closure and representation theoretic decomposition for constructing explicit analytic representations of the non-trivial zeros of the Riemann zeta function. We first construct a finitely generated differential algebraic closure space K for the ζ function and its associated derivations, and realize it as a reproducing kernel Hilbert space H via weighted Sobolev norms. By defining spectral projection operators Pn and a specific continuous linear functional Λ on this space, we prove that the imaginary parts of the zeros can be explicitly represented as tn = Λ(Pnψ0) + o(1).Starting from this representation and combining asymptotic analysis with classical theory, we derive the leading asymptotic formula tn ∼ 2πn/(logn − A) and provide a theoretical derivation of the global constant A =1 +γ −log(4π). Further, by analyzing the variational structure of representation-theoretic components, we introduce a local scale invariant Rn and obtain the structural form of a complete second-order asymptotic formula: tn = (2πn)/(logn − A)1+(αRn +β)/n+o(1/n), where the coefficients α,β are determined by the representation-theoretic structure of the closure. We provide a systematic numerical verification scheme,including a high-precision recursive algorithm based on theoretical corrections, convergence basin dynamics analysis, and preliminary validation of extending the framework to Dirichlet L-functions, with results supporting the predicted form. This framework transforms the zero localization problem into a representation-theoretic problem within a differential algebraic closure, providing a new perspective and tool for understanding zero distribution, exploring the Hilbert-P´olya approach, and studying zeros of more general L-functions.
shifa liu (Wed,) studied this question.