Following off of @user64494, the Mathematica code
Product[(1/2 + n)^(1/2 + n)/(Exp[1]*n)^(2*1/2)/(n - 1/2)^(n - 1/2)*(7/12 + n)^(7/12 + n)/(Exp[1]*n)^(2*7/12)/(n - 7/12)^(n - 7/12)*(1/20 + n)^(1/20 + n)/(Exp[1]*n)^(2*1/20)/(n - 1/20)^(n - 1/20)*(13/20 + n)^(13/20 + n)/(Exp[1]*n)^(2*13/20)/(n - 13/20)^(n - 13/20)/((1/4 + n)^(1/4 + n)/(Exp[1]*n)^(2*1/4)/(n - 1/4)^(n - 1/4))/((1/12 + n)^(1/12 + n)/(Exp[1]* n)^(2*1/12)/(n - 1/12)^(n - 1/12))/((11/20 + n)^(11/20 + n)/(Exp[1]*n)^(2*11/20)/(n - 11/20)^(n - 11/20))/((3/20 + n)^(3/20 + n)/(Exp[1]*n)^(2*3/20)/(n - 3/20)^(n - 3/20)), {n, 1, Infinity}]
gives the closed form output
$$\frac{{2}^{1/4} 3^{13/20} 11^{11/20} 5^{\frac{1}{5} \zeta \left(-1,-\frac{1}{4}\right)-\frac{1}{480}} \exp \left(\frac{1}{5} \zeta ^{(1,0)}\left(-1,-\frac{1}{4}\right)+\zeta ^{(1,0)}\left(-1,\frac{19}{20}\right)-\zeta ^{(1,0)}\left(-1,-\frac{1}{20}\right)+\frac{97 C}{60 \pi }+\frac{361}{480}\right)}{7^{7/12} 13^{13/20} \pi ^{3/4} {Glaisher}^{1/40}}$$$$\frac{{2}^{1/4} 3^{13/20} 11^{11/20} 5^{\frac{1}{5} \zeta \left(-1,-\frac{1}{4}\right)-\frac{1}{480}} \exp \left(\frac{1}{5} \zeta ^{(1,0)}\left(-1,-\frac{1}{4}\right)+\zeta ^{(1,0)}\left(-1,\frac{19}{20}\right)-\zeta ^{(1,0)}\left(-1,-\frac{1}{20}\right)+\frac{97 C}{60 \pi }+\frac{361}{480}\right)}{7^{7/12} 13^{13/20} \pi ^{3/4} {A}^{1/40}}$$
where $A$ is the Glaisher's constant which is approximately 1.28243 and $\zeta$ is the (generalized) Riemann zeta function. Evaluating the above mess to 20 decimal places gives
$$0.780~459~197~412~937~486~21.$$
