= Solution
Interpret the supplied <free energies> as standard reaction <free energies> and use $c^\circ=1\,\mathrm{mol\,L^{-1}}$. The dimensionless <equilibrium constants> are
$$
K_C=\exp\!\left(-\frac{15000}{RT}\right)=2.3477\times10^{-3},
\qquad K_D=\exp\!\left(\frac{35000}{RT}\right)=1.3651\times10^6.
$$
Under the ideal-solution approximation, $[C]=K_C[L]$ and $[D]=(K_D/c^\circ)[L]^2$. Set $k_D=K_D/c^\circ$, with units $\mathrm{M^{-1}}$. Conservation counts two original <molecules> in each <dimer>:
$$
c_0=[L]+[C]+2[D]=(1+K_C)[L]+2k_D[L]^2.
$$
The physical, positive root can be written without subtractive cancellation as
$$
\boxed{[L]=\frac{2c_0}{1+K_C+\sqrt{(1+K_C)^2+8k_Dc_0}}}.
$$
The resulting <concentrations> are:
|| Initial analytical <concentration>
|| $[L]$/M
|| $[C]$/M
|| $[D]$/M
| $0.100\,\mathrm M$
| \b[$1.9120\times10^{-4}$]
| \b[$4.4888\times10^{-7}$]
| \b[$0.0499042$]
| $1.000\,\mathrm M$
| \b[$6.0502\times10^{-4}$]
| \b[$1.4204\times10^{-6}$]
| \b[$0.499697$]
For example, the first row satisfies $[L]+[C]+2[D]=0.100\,\mathrm M$; using $[L]+[C]+[D]$ would incorrectly count <dimers>. At both <concentrations> almost all molecular units belong to <dimers>.
<dimer> formation is favored by sufficiently high <concentration> because it joins two <molecules>. Dilution favors the separated species, whereas $[C]/[L]=K_C$ is concentration-independent in this ideal model. More than half the molecular units are dimerized when $2[D]>[L]+[C]$, equivalently $c_0>(1+K_C)^2/k_D\simeq7.36\times10^{-7}\,\mathrm M$. The supplied <free energies> at one <temperature> do not determine a <temperature> trend without additional <enthalpy> information. At $1\,\mathrm M$, neglected nonideal activities could be significant; the table is the requested concentration-based model.
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