A Unified Framework of Hypergeometric Rapidly Convergent Series for L-Functions and Generalized L-Functions: From the Lerch Transcendent to Systematic Study

Based on the Mellin-Barnes integral representation of the Lerch transcendent and Olde Daalhuis's terminating asymptotics, this paper establishes a unified mathematical framework for deriving hypergeometric rapidly convergent series representations of L-functions and generalized L-functions. We systematically analyze the arithmetic properties of Dirichlet L-functions, Hecke L-functions, Artin L-functions, elliptic curve L-functions, automorphic L-functions, and more general L-functions, revealing that all of them can be expressed as values or derivatives of generalized hypergeometric functions at rational points. Six equivalent forms are established: the Lerch transcendent form, the Bernoulli series form, the conjugate Bernoulli form, the hypergeometric series form, the Olde Daalhuis asymptotic form, and the Banerjee-Wilkerson Lambert series form, with rigorous transformation proofs. A unified generating function L s, ;a, b;c;z) =₍=₀^ (a) ₙ (b) ₙ (c) ₙ (2n) ! (n!) ²L (s+n, ) zⁿ2^{2n} is constructed, where L (s, ) denotes Dirichlet L-functions, and all L-function values can be represented by this generating function. The convergence rates of the six forms are systematically compared, computational complexities are analyzed, and optimal parameter selection criteria are given. The framework is extended to multiple L-functions and p-adic L-functions, with corrected convergence conditions. Complete numerical verification code and computational results are provided. All results are presented with complete derivations and rigorous mathematical proofs.