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) —>-i5.5pi
W-3((20.,-1.E400)): (914.215721783,-20.3980438811)—>-i6.5pi
W-3((20.,9.E490)): (1123.43965171,-17.2633942517) —>-i5. 5pi
W-3((20.,-1.E490)): (1121.24433569,-20.4021582594) —>-i6.5pi
W0((20.,1.E490)):(1121.24450008,1.56939663599) —>i1/2pi