What is the mathematical expression of the Lambda-CDM model? The conceptual meaning of this model is clear to me. But where can I find the actual mathematical model? 
The Wikipedia page says "The simple ΛCDM model is based on six parameters". Six parameter where? in what equation do you adjust this parameters? how does this model looks like? 
When scientist work with the ΛCDM model, what do they actually work with?
 A: A short answer to this question might be as follows:
Einstein's equations of general relativity for a form of the geometry and a special mixture of matter components. This yields the evolution of the universe on large scales after the time of inflation and reheating upto today.
The geometry part is usually referred to Friedmann-Robertson-Walker geometry (FRW), which is in the simplest case an expanding three dimenional space. The metric for the four-dimensional spacetime is given by
$$ds^2 = -dt^2+a(t)^2 (dx^2+dy^2+dz^2) = -dt^2 +a(t)^2(dr^2+r^2d\theta^2+\sin^2\theta d\phi^2)$$
The first version uses regular cartesian coordinates. The second version shows the three-dimensional spatial geometry in polar coordinates, which reflects the assumption of homogeneity and isotropy on large scales.
The only unkown function is $a(t)$ the scale factor of the universe that basically tells you, how fast the universe is expanding. This depends on the matter content of the universe.
In the simplest version, one assumes that the matter of the universe behaves like a perfect fluid. The energy-momentum tensor only has diagonal entries. The energy density $\rho$ und the pressure $p$, yielding $T_{^\mu_\nu}=diag(\rho,p,p,p)$.
Einstein's field equations reduce to the very famous Friedman equation:
$$ \left(\frac{\dot a }{a}\right)^2= \frac{8\pi G}{3} \rho$$.
For dark-energy domination it follows that $a(t)\propto \exp(t)$.
For dust like matter $a(t)\propto t^{2/3}$ and for 
radiation like matter $a(t) \propto t^{1/2}$.
All three stages have been present in the evolution of our universe, since the energy densities dilute differently with the expanding universe, radiation domination was initiallly. It has been diluted with $a^4$. It was followed by a stage of matter domination (diluted with $a^3$), which has only recently been replaced by a stage of dark energy domination (diluted with $a^0$).
$\Lambda$ stands for the dark energy, nowadays contributing 70 percent of the energy density.
The remaining 30 percent are mostly provided by cold dark matter (CDM). These are two of the six parameters. They are accompanied by the energy density of radition, todays expansion rate $H_0$, the baryonic energy density, and the total amount of energy density, which is usually assumed to be critical, which justifies the chosen geometry at the beginning.
A: From your link:

The simple ΛCDM model is based on six parameters: physical baryon density parameter; physical dark matter density parameter; the age of the universe; scalar spectral index; curvature fluctuation amplitude; and reionization optical depth.[23] In accordance with Occam's razor, six is the smallest number of parameters needed to give an acceptable fit to current observations; other possible parameters are fixed at "natural" values, e.g. total density parameter = 1.00, dark energy equation of state = −1. (See below for extended models that allow these to vary.)
The values of these six parameters are mostly not predicted by current theory (though, ideally, they may be related by a future "Theory of Everything"), except that most versions of cosmic inflation predict the scalar spectral index should be slightly smaller than 1, consistent with the estimated value 0.96. The parameter values, and uncertainties, are estimated using large computer searches to locate the region of parameter space providing an acceptable match to cosmological observations. From these six parameters, the other model values, such as the Hubble constant and the dark energy density, can be readily calculated.

You ask:

The Wikipedia page says "The simple ΛCDM model is based on six parameters". Six parameter where? in what equation do you adjust this parameters? how does this model looks like?

The parameters are "measured" i.e. estimated by observational data, not parameters for adjustment.
This is an article by NASA that has a great number of references,  and this   review : may be useful for people wanting to delve further on how people work numerically with the model.

The homogeneous, isotropic, and flat ΛCDM universe favored by observations of the cosmic microwave background can be described using only Euclidean geometry, locally correct Newtonian mechanics, and the basic postulates of special and general relativity. We present simple derivations of the most useful equations connecting astronomical observables (redshift, flux density, angular diameter, brightness, local space density,...) with the corresponding intrinsic properties of distant sources (lookback time, distance, spectral luminosity, linear size, specific intensity, source counts,...). We also present an analytic equation for lookback time that is accurate within 0.1% for all redshifts z. The exact equation for comoving distance is an elliptic integral that must be evaluated numerically, but we found a simple approximation with errors <0.2% for all redshifts up to z≈50.

