In the functions I1 and I2 your are using {} where I think () is needed, you don't need to generate a list. Your functions can be evaluated for a list of values of Gamma if you rewrite your functions to take Gamma as an input. That way you can use a list of Gamma functions directly in the plot function.

Also I don't think this plot needs 10000 PlotPoints to show all the details you need.

Ug = 1/2; U = -1/12; u = 1/4; s = -13; w = 10; p = -5;
\[CapitalGamma] = {0.001, .1, 1., 2, 4, 6, 8, 10, 20};
e = 2*s + p + Ug + w;

Em1[y_] := s + p + U + y;
Em2[y_] := s + p + u + y;

Ef[x_, y_] := 2*s + x + y;

I1[y_, \[CapitalGamma]_] := 
  1/((Em1[y] - e)^2 + (Pi*\[CapitalGamma])^2);
I2[y_, \[CapitalGamma]_] := 
  1/((Em2[y] - e)^2 + (Pi*\[CapitalGamma])^2);

S[y_, \[CapitalGamma]_] := (I1[
       y, \[CapitalGamma]]/(I1[y, \[CapitalGamma]] + 
        2*I2[y, \[CapitalGamma]]))*
    Log[2, (I1[y, \[CapitalGamma]] + 2*I2[y, \[CapitalGamma]])/
      I1[y, \[CapitalGamma]]] + 
   2*I2[y, \[CapitalGamma]]/(I1[y, \[CapitalGamma]] + 
       2*I2[y, \[CapitalGamma]])*
    Log[2, (I1[y, \[CapitalGamma]] + 2*I2[y, \[CapitalGamma]])/(2*
        I2[y, \[CapitalGamma]])];

Plot[Evaluate@S[y, \[CapitalGamma]], {y, -10, 5}, PlotRange -> All, 
 AxesLabel -> Automatic, 
 GridLines -> {{-2.75, -2.55, -2.41, -1.61}, {0}}, 
 GridLinesStyle -> Red, PlotLegends -> \[CapitalGamma]]

Hi Ryo,

E is a built-in symbol representing the mathematical constant e. It cannot be assigned a value

E := 2*s + p + Ug + w;

Here is one way to do this

e = 2*s + p + Ug + w; (* Use e, not E *)

plots = Table[
  Plot[S[y], {y, -10, 5},
   PlotRange -> All,
   AxesLabel -> Automatic,
   PlotLabel -> "Γ" <> " = " <> ToString@Γ,
   PlotLabels -> "Γ" <> " = " <> ToString@Γ,
   GridLines -> {{-2.75, -2.55, -2.41, -1.61}, {0}},
   GridLinesStyle -> Red],
  {Γ, {0.001, .1, 1., 2, 4, 6, 8, 10, 20}}];

Single plot

Show[plots, PlotRange -> All, PlotLabel -> None]

Multiple plots

plots // Partition[#, UpTo@3] & // Grid[#, Frame -> All] &

You can change the specific values for Γ by changing this list

{0.001, .1, 1., 2, 4, 6, 8, 10, 20}