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 Λ_k(ω) = Σ_{n=1}^∞ (1/α_{-k}(nω) - 1) at roots of unity ω, where α_{-k}(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 Λ_k(ω̅) = Λ_k(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.19025728[2] Hussain, M.R. (2026). Quantum Modularity of Mock Theta Functions from the Alpha Hierarchy. Zenodo. DOI:10.5281/zenodo.19038843