1. More generally in $d$ spacetime dimensions, the perturbative gravity Lagrangian density is schematically of the form$^1$
$$  {\cal L}~=~\frac{\sqrt{|g|}}{16\pi G}\frac{f(\ell_{\!P}^2 R)}{\ell_{\!P}^2}~=~(\partial h)^2+\frac{\hbar}{16\pi\ell_{\!P}^d}f_{\rm int}(\ell_P\partial,\kappa h),\tag{1}$$
where the metric field
$$g_{\mu\nu}~=~\eta_{\mu\nu}+\kappa h_{\mu\nu},\qquad \kappa~:=~2\sqrt{8\pi G}, \tag{2}$$
is expanded around a background $\eta_{\mu\nu}$, typically the Minkowski spacetime. Here $R$ is a shorthand for either the Riemann, Ricci or scalar curvature tensor. The functions$^2$ in eq. (1) take dimensionless arguments into dimensionless values. The Planck length $\ell_{\!P}$ satisfies
$$ \hbar G~=~\ell_{\!P}^{d-2},\tag{3}$$
cf. e.g. Ref. 3.

2. For dimensional reasons, The graviton-graviton [vacuum-polarization](https://en.wikipedia.org/wiki/Vacuum_polarization)/[self-energy](https://en.wikipedia.org/wiki/Self-energy) $\widetilde{\Pi}$ [which consists of  amputated Feynman diagrams stripped of a delta function $\delta^d(k_1\!+\!k_2)$ of total wave number conservation] has dimension
$$\begin{align} [\widetilde{\Pi}]~=~&\text{(Angular momentum)}\times\text{Length}^{-2}\cr
~=~&\frac{\text{Energy}^{2}}{\text{Angular momentum}},\end{align}\tag{4}$$
cf. e.g. [this](https://physics.stackexchange.com/q/538319/2451) Phys.SE post.
Therefore $\widetilde{\Pi}$ is of the form
$$\widetilde{\Pi}(E)~=~\frac{E^2}{\hbar}f\left(\frac{E}{m_P}\right),\tag{5}$$
where $E$ is the CM energy, and 
$$m_P~=~\frac{\hbar}{\ell_{\!P}}~\stackrel{(3)}{=}~\hbar(\hbar G)^{-1/(d-2)}\tag{6}$$
is the Planck mass.

3. For $d=4$ the variable
$$\frac{E}{m_P}~=~E\sqrt{\frac{G}{\hbar}}\tag{7}$$
in eq. (5) leads to the perturbative series$^3$ mentioned in Refs. 1 & 2.

References:

1. S. Weinberg, _UV divergences of quantum theories of gravitation._ Published in _GR: An Einstein centenary survey_ (Eds. S.W. Hawking & W. Israel), 1979; p.797.

2. A. Zee, _QFT in a nutshell,_ 2010; section III.2 p.169-172 & section VIII.1 p.434-435.

3. B. Zwiebach, _A first course in String Theory,_ 2nd edition, 2009; section 3.8, eq. (3.108).

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$^1$ We work in units where the speed of light $c=1$ is one.

$^2$ We call all such functions $f$, even though they are different.

$^3$ Here we are implicitly assuming that each Feynman diagram with an even/odd number of external legs only comes with an even/odd power of the coupling constant $\kappa$, respectively. This is true in [GR](https://en.wikipedia.org/wiki/General_relativity).