  We examine a family ${}_pG_{q}^{\mathbb
C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$ of integrals of Mellin-Barnes type
over the space ${\mathbb Z}\times {\mathbb R}$, such functions $G$ naturally
arise in representation theory of the Lorentz group. We express
${}_pG_{q}^{\mathbb C}(z)$ as quadratic expressions in the generalized
hypergeometric functions ${}_{p}F_{q-1}$ and discuss further properties of the
functions ${}_pG_{q}^{\mathbb C}(z)$.
