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

[Gil]

Some "guessed" results for input z =a+ib, with a = -INF:

W-3(-INF +i20) = inf-i(5pi)
W-3(-INF - i20) = inf-i(7pi)

W-2(-INF +i20) = inf-i(3pi)
W-2(-INF - i20) = inf-i(5pi)

W-1(-INF +i20) = inf-i(pi)
W-1(-INF - i20) = inf-i(3pi)


W0(-INF +i20) = inf+i(pi)
W0(-INF - i20) = inf-i(pi)

W1(-INF + i20) = inf+i(5pi)
W1(-INF - i20) = inf+i(3pi)

W2(-INF + i20) = inf+i(5pi)
W2(-INF - i20) = inf+i(pi)

W3(-INF + i20) = inf+i(7pi)
W3(-INF - i20) = inf+i(5pi)

Wk(-INF, +ib, b>0) = inf+i(2k+1)(pi)
Wk(-INF, +ib, b<0) = inf+i(2k+1-2)(pi)

Wk(-INF, +ib, b>0) = inf+i(2k+1)(pi)
Wk(-INF, +ib, b<0) = inf+i(2k-1)(pi)

Conclusion
Wk(-INF, +ib, b≠0) = inf+i(2k+sign(b))(pi)
& we may check Wk(-INF, +ib, b=0) = inf+i(2k+1)(pi)

Summary
Wk(-INF, +ib) =inf+i(pi) *(2k+1, if b>=0; 2k-1, if b<0)