Lambert function and Wolfram or "±infinity+i×K=±infinity" ?

[Gil]

For k=0 and z=20+i×1E490,
I find the approached result you mentioned.

However, for k=-3, my HP50G program gives quite different multiple values of pi for the imaginary part.

See my results:
W-3((20.,1.E400)): (914.215792373,-17.2598823971)
-17.26i = about -i5.5pi or iPI(2k+0.5),
according to the formula in a previous post

W-3((20.,-1.E400)): (914.215721783,-20.3980438811)
-20.40i = about -i6.5pi or iPI(2k-0.5),
according to the formula in a previous post

W-3((20.,9.E490)): (1123.43965171,-17.2633942517)
-17.26 pi = about -i5.5pi

W-3((20.,-1.E490)): (1121.24433569,-20.4021582594)
-20.40i = about -i6.5pi

W0((20.,1.E490)):(1121.24450008,1.56939663599)
1.57pi = about 1/2pi

My result for your case with k=-3 correspond to the snapshot included here taken from Wolfram Alpha.

Have you a way to check again your result?