One formula involving the gamma function is the Gauss multiplication formula:

Γ(z)Γ(z+1/n)...Γ(z+(n-1)/n)=τ^[(n-1)/2]*n^(1/2-nz)*Γ(nz)

Note that this formula is simpler with tau than pi.

The special case of this formula when n=2 is Γ(z)Γ(z+1/2)=sqrt(τ)*2^(1/2-2z)*Γ(2z)

Now, by setting z=1/2, we get Γ(1/2)Γ(1)=sqrt(τ)*2^(-1/2)*Γ(1)

Since 0!=1, Γ(1)=1, and we simplify the equation to get

Γ(1/2)=sqrt(τ)/sqrt(2)=sqrt(π)

So the only reason that Γ(1/2) is the square root of pi is because we used a special case of a more general formula which is simpler with tau, but in this special case, a factor of two happens to cancel out with the 2 in 2π, so the pi really is half-tau there. The constant that simplifies the general formula is more natural, not the constant that simplifies one case of it due to factors of 2 cancelling out.