L=200;λ=0.05;Ω=0.03;σ=0.2;LaunchKernels[];ClearAll[I0,I1];I0[a_?NumericQ,m_?NumericQ]:=4/Sqrt[4πm]*NIntegrate[Exp[-(τ^2)/(2σ^2)]*Cos[Ωτ+(2πm)/(La)Sinh[aτ]]*Cos[(2πm)/(La)(Cosh[aτ]-1)],{τ,0,Infinity},AccuracyGoal6,Method"MultiPeriodic",PrecisionGoal6,MaxRecursion12];I1[a_?NumericQ,m_?NumericQ]:=4/Sqrt[4πm]*NIntegrate[Exp[-(τ^2)/(2σ^2)]*Cos[-Ωτ+(2πm)/(La)Sinh[aτ]]*Cos[(2πm)/(La)(Cosh[aτ]-1)],{τ,0,Infinity},AccuracyGoal6,Method"MultiPeriodic",PrecisionGoal6,MaxRecursion12];η0[a_?NumericQ]:=λ*Sum[I0[a,m],{m,1,10000}];η1[a_?NumericQ]:=λ*Sum[I1[a,m],{m,1,10000}];F[a_?NumericQ]:=(1-η0[a]η1[a])/Sqrt[(1+η0[a]^2)(1+η1[a]^2)]ListPlot[ParallelTable[{a,F[a]},{a,1,50,1}],FrameTrue,FrameStyleDirective[Black,12,Bold],FrameLabel{Style["a",Italic,13,Bold,Black],Style["F(a)",Italic,13,Bold,Black]},DataRange{1,50},PlotRangePadding0]