New Lambda functions at the roots of unity with special values and infinite series.

This is Part 3 of 1. We study the Lambda functions Λₖ (ω) = Σ₍=₁^∞ (1/α-₊ (nω) - 1) at roots of unity ω, where α-₊ (z) are the alpha functions introduced in 1. For k=2 and k=3 we compute numerical values at ω = 1, -1, i, -i, e^2πi/3, e^4πi/3 using high-precision summation. Conjugation symmetry Λₖ (ω̅) = Λₖ (ω) is observed in all cases. Exact closed forms are given for ω = i: Λ₂ (i) = 1 + ψ⁽¹⁾ (-i) Re Λ₂ (i) = 1/2 − π²/ (2 sinh² (π) ) Im Λ₂ (i) = −Im ψ⁽¹⁾ (1+i) Λ₃ (i) = (3i/8) (ψ (1-i) −ψ (1-3i) ) + (3/4) ζ (2, 1-i) − (i/2) ζ (3, 1-i) where ψ is the digamma function, ψ⁽¹⁾ the trigamma function, and ζ (s, z) the Hurwitz zeta function. These formulas match direct summation within 10⁻⁵ for Λ₂ (i) and 4×10⁻⁸ for Λ₃ (i). Partial results for Λ₄ (1) are also given. Conjectures are stated for general k and other roots of unity. 1 Hussain, M. R. (2026). Alpha Functions: A New Hierarchy with Connections to Elliptic Integrals and Mock Theta Functions. Zenodo. DOI: 10. 5281/zenodo. 190257282 Hussain, M. R. (2026). Quantum Modularity of Mock Theta Functions from the Alpha Hierarchy. Zenodo. DOI: 10. 5281/zenodo. 19038843