IFFT for exponential function

So the inverse Fourier transform of 1/(1+i*s) is exp(-y), I'm trying to verify this using ifft. My code is below and note that I use factor to account for the necessary N/(2pi) prefactor. The shape is normally correct but the amplitude is always off. I use s to denote frequency space and y for the regular function domain. I don't have a massive background on signal processing but if I'm missing something major I can read into it.

NN=2^16;

dy=.01;

ds=2*pi/(dy*NN);

grid=0:ds:(NN-1)*ds;

Lt=(1./(1+1i*20.*grid));

factor=ds*NN/(2*pi);

transform=ifft(Lt)*factor;

freq=0:pi/ds:2*pi/((NN-1)*ds);

density=real(transform);

plot(freq(1:length(freq)),density(1:NN/2));