As I know, for Hydrogen atom a $1s$ Slater Type Orbital (STO) equation is (I can get it from here):
\begin{equation}\label{STO} \mathrm{STO} = \sqrt{\frac{\zeta^3}{\pi}} e^{-\zeta r}. \end{equation}
With remark from Szabo A., Ostlund N.S. Modern Quantum Chemistry: Intro to Advanced Electronic Structure Theory, Dover, 1996:
For example, the standard exponent for the $1s$ basis function of hydrogen is , $\zeta = 1.24$. This is larger than the $\zeta = 1.00$ exponent of the hydrogen atom, since the hydrogen $1s$ orbital in average molecules is known to be "smaller" or "denser" than in the atom.
Gaussian Type Orbital (GTO) equation for $1s$-orbital (where did this coefficient $\left(\frac{2\alpha}{\pi}\right)^{3/4}$ come from?):
\begin{equation} \mathrm{GTO} = \left(\frac{2\alpha}{\pi}\right)^{3/4} e^{-\alpha r^2} \end{equation}
So, I have good analitical form of Hydrogen STO and GTO, and using formula \begin{equation}\label{} \sqrt{\frac{\zeta^3}{\pi}}e^{-\zeta r} \approx C_1 \left(\frac{2\alpha_1}{\pi}\right)^{3/4}e^{-\alpha_1 r^2} + C_2 \left(\frac{2\alpha_2}{\pi}\right)^{3/4}e^{-\alpha_2 r^2} + C_3 \left(\frac{2\alpha_3}{\pi}\right)^{3/4}e^{-\alpha_2 r^2}. \end{equation} and copyng $\alpha_i$'s and $C_i$ from basissetexchange.org
#-----------------------------------------------------
# Basis Set Exchange
# Version v0.8.13
# https://www.basissetexchange.org
#-----------------------------------------------------
# Basis set: STO-3G
# Description: STO-3G Minimal Basis (3 functions/AO)
# Role: orbital
# Version: 1 (Data from Gaussian09)
#-----------------------------------------------------
BASIS "ao basis" PRINT
#BASIS SET: (3s) -> [1s]
H S
0.3425250914E+01 0.1543289673E+00
0.6239137298E+00 0.5353281423E+00
0.1688554040E+00 0.4446345422E+00
END
I can plot Hydrogen orbital
Now I want to do the same for a Lithium atom using data
BASIS "ao basis" PRINT
#BASIS SET: (6s,3p) -> [2s,1p]
Li S
0.1611957475E+02 0.1543289673E+00
0.2936200663E+01 0.5353281423E+00
0.7946504870E+00 0.4446345422E+00
Li SP
0.6362897469E+00 -0.9996722919E-01 0.1559162750E+00
0.1478600533E+00 0.3995128261E+00 0.6076837186E+00
0.4808867840E-01 0.7001154689E+00 0.3919573931E+00
END
But, I do not understand where I can get a analitical formula for GTO's primitives. All of references, I saw:
\begin{equation}\label{GTO}
\mathrm{GTO}(x,y,z;\alpha,\ell_1,\ell_2,\ell_3) = N x^{\ell_1} y^{\ell_2} z^{\ell_3} e^{-\alpha r^2}
\end{equation}
with unknown normalizing factor $N$. Sometimes I saw for $1p$-orbital
\begin{equation}
\mathrm{GTO}(p_x) = \left(\frac{128\alpha^5}{\pi^3}\right)^{1/4} x \exp(-\alpha r^2)
\end{equation}
but I don't understand where the factor $\left(\frac{128\alpha^5}{\pi^3}\right) $ comes from.
So, my general question: where can I see the analytical view of the GTO primitive (like for STO's), and how can you find $\zeta$ for atom obital? (how to read such data?)