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We study the meromorphic function $ D(s) := -\frac{\zeta'(s)}{s\zeta(s)} - \frac{1}{s-1}, $ obtained by removing the polar part of $ -\zeta'(s)/(s\zeta(s)) $ at $ s = 1 $. The central observation is the Laplace representation $ D(s) = \int_{0}^{\infty} (\psi(e^t) - e^t) e^{-st}\, dt, \quad \operatorname{Re}(s) > 1, $ which identifies $ D(s) $ as the Laplace transform of the prime-counting error $ E(t) := \psi(e^t) - e^t $. Six consequences follow, all proved unconditionally: (I) Complete meromorphic structure of $ D $ on $ \mathbb{C} $: poles at non-trivial zeros $ \rho $ (residue $ -1/\rho $), trivial zeros $ -2k $ (residue $ 1/(2k) $), and $ s = 0 $ (residue $ -\log 2\pi $); removable singularity at $ s = 1 $ with $ D(1) = -1 - \gamma_0 $. (II) The Laplace transform converges absolutely for $ \operatorname{Re}(s) > 0 $ unconditionally and for $ \operatorname{Re}(s) > 1/2 $ under RH; the abscissa of absolute convergence equals $ \sup_{\rho} \operatorname{Re}(\rho) $. (III) $ D(1+i) \in L^2(\mathbb{R}) $ and $ \frac{1}{2\pi} \int_{-\infty}^{\infty} |D(1+it)|^2\, dt = \int_{1}^{\infty} \frac{(\psi(x)-x)^2}{x^3}\, dx < \infty $ unconditionally. By contrast, $ (-\zeta'/\zeta)(1+i) \notin L^2(\mathbb{R}) $. (IV) Under RH, the right-hand side expands into an explicit, absolutely convergent six-term sum indexed by zeros and constants. (V) Unconditionally, $ \int_{1}^{\infty} (\psi(x)-x)x^{-2}\, dx = -1 - \gamma_0 $ (Newman's theorem, Theorem 1.7). (VI) $ D $ extends holomorphically to $ \operatorname{Re}(s) > 1/2 $ if and only if the Riemann Hypothesis holds; more precisely, the abscissa of holomorphy of $ D $ equals $ \sup_{\rho} \operatorname{Re}(\rho) $. The paper also includes an independent derivation of $ \sum_{\rho} \operatorname{Re}(1/\rho) = \log(2\pi) - 1 $ and extends all results to Dirichlet $ L $-functions. Note on corrections. A prior version contained four sign errors: in the residue of $ D $ at $ s = 0 $; in the explicit formula for $ \psi(x) - x $; in the zero-sum formula; and in the cross-terms $ I_{AL} $, $ I_{AC} $, $ I_{LC} $ of Theorem 1.5. All are corrected here. Recommended Comments field (optional but very useful): 18 pages. Corrected and expanded version (v2). Primary 2020 MSC: 11M06; Secondary: 11N05, 44A10, 11M26. Submitted to Proc. Amer. Math. Soc. Independent researcher.