Find a hypergeometric formula embracing three specific cases

Question

For a parameter value $a=\frac{1}{4}$, I have the result \begin{equation} Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}{8};k+\frac{19}{8},k+\frac{23}{8};1\right)}{\Gamma \left(k+\frac{19}{8}\right) \Gamma \left(k+\frac{23}{8}\right)}. \end{equation} Further, for $a=\frac{1}{2}$, there is the result \begin{equation} Q(k,\frac{1}{2})=\frac{\Gamma \left(2 k+\frac{9}{2}\right)}{\sqrt{\pi } \Gamma (2 k+5)} \end{equation} and for $a=1$, \begin{equation} Q(k,1)=\frac{4^{k+3} \Gamma \left(k+\frac{7}{2}\right)^2 \Gamma \left(k+\frac{9}{2}\right)}{\pi \Gamma (k+5) \Gamma \left(2 k+\frac{13}{2}\right)}. \end{equation} Can a common formula $Q(k,a)$ be found to embrace these three cases? (I have many more results for larger half-integer and integer values of $a$, that hopefully such a formula would encompass too, but will not now present them.)