<s>[INST] Hypergeometric value Is their a closed form for the following 
$${}_2F_1 \left(a,b;c;\frac{1}{2} \right)$$
I would use  the following 
$${}_2F_1 \left(a,b;c;x \right)= \frac{\Gamma(c)}{\Gamma(c-b)\Gamma(b)} \int^1_0 t^{b-1}(1-t)^{c-b-1} (1-xt)^{-a} \, dt $$
But it wasn't a success !
Edit: Corrected integral representation (swapped arguments in $\Gamma$  fraction)
 [/INST] There are closed forms for the 2F1 but their exact form depend on the values and the relations between the parameters.


*

*The trivial case is if $a$ or $b$ is a negative integer: the function becomes a finite sum of rational expressions in the parameters. if you choose $a=-4$ and if you name $d$ the value you have fixed at $1/2$, you got
$ 1 - \frac{4 b d}{c} +\frac{6 b (b+1) d^2}{c (c+1)} - \frac{4 b (b+1) (b+2) d^3}{c (c+1)
   (c+2)} - \frac{(-b-3) b (b+1) (b+2) d^4}{c (c+1) (c+2) (c+3)}$
and setting $d = 1/2$:
$ 1 -\frac{2 b}{c} + \frac{3 (b+1) b}{2 c (c+1)} - \frac{(b+1) (b+2) b}{2 c (c+1)
   (c+2)} - \frac{(-b-3) (b+1) (b+2) b}{16 c (c+1) (c+2) (c+3)} $


*

*if $a$ or $b$ is equal to $c$, they cancel each other, giving a simpler $_1F_0$.
$_2F_1( a, b ; b ; d)  =  _1F_0(a; d) = (1-d)^{-a} $

*for small rationals, there are more interesting closed forms involving elementary functions and elliptic functions and their inverses. Among classical cases
$_1F_0(1/2 ; d) = \frac{1}{\sqrt{1- d}} $
$_2F_1(1/2, 1/2 ; 3/2; d) = \frac{1}{\sqrt d} \sin^{-1} (\sqrt d )$
$_2F_1(1/2, 1 ; 3/2; d) = \frac{1}{\sqrt d} \tanh^{-1} (\sqrt d )$
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