Meaning of transfer function in s evaluated at a complex number
The "real world" is along the \$j\omega\$ axis and, ignoring such things as negative frequencies, measurable reality is from the origin and upwards (frequency rising). The pole zero diagram vertical axis embodies the bode plot amplitude like this: -



The above (and below) is just an example for a 2nd order low pass filter. 
Looking vertically down on the diagram above gives the traditional pole-zero view: -



Picture source.

Basically it's meaningless to analyse \$H\$ in any other place than the axis that embodies the bode plot.