The main purpose of this paper is to compute all irreducible spherical functions on $G={ SL}(2,{ \mathbb C})$ of arbitrary type $\delta\in \hat K$, where $K={ SU}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the three dimensional hyperbolic space ${ \mathbb H}=G/K$. The entries of $H$ are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of $H$ in terms of Gauss' functions ${ }_2F_1$. Just as in the compact instance treated in [GPT] there is a useful role for a special class of generalized hypergeometric functions ${ }_{ p+1}F_p$.
Categories: math.RT math.CA