This paper comprehensively extends the theory of closed-form expressions and systematic theory for complex-order Riemann zeta functions to alternating sum forms of Riemann-type functions (exemplified by the Dirichlet eta function) and their general cases (Dirichlet L-functions). Through systematic application of complex function theory, holomorphic continuation of the Gamma function, complex fractional calculus, and analytic theory of generalized hypergeometric functions,we construct a unified representation framework applicable to arbitrary complex numbers in the complex plane except for possible isolated singular points.First, we rigorously generalize integral closed-form expressions applicable to right half-planes to the entire complex plane via carefully designed complex contour integral techniques, providing a universal expression based on contour integration, and detailing its domain of convergence and analytic structure. Second, we establish representations based on complex-order hypergeometric functions, clarifying the analytic properties and series expansions of principal terms and correction terms throughout the complex plane. Furthermore, we develop Gamma-rational approximation theory applicable to complex domains and provide complex error estimates based on the maximummodulus principle. Particularly, we study properties of function values when arguments take complex algebraic numbers, providing upper bounds for irrationality measures. Finally, we establish complex-order fractional differential equations, revealing the differential structure of such functions on the complex plane.This paper further deepens five core directions: (1) Closed-form expressions for derivatives of all orders on the complex plane; (2) Study of non-trivial zero properties based on complex integral representations and complex differential equations; (3) Systematic extension of the theory to complex-order multiple L-functions and complex multiple polylogarithms; (4) Development of high-precision algorithms such as adaptive complex contour integration and accelerated complex hypergeometric series; (5) Extension of applications in areas such as complex-dimensional regularization in quantum field theory and complex fractional-order signal processing systems.All theoretical results are accompanied by rigorous complex-analytic proofs, detailed derivations, and numerical verification based on interval arithmetic, providing systematic theoretical,algorithmic, and application tools for studying complex-order alternating sum zeta functions and related functions.
shifa liu (Wed,) studied this question.