Variable $s in searchform.php

In the searchform.php of the default template and in many other templates, the search request is returned to the search form with the variable $s.

I would like to change its value within a plugin, but I couldn’t seem to find out where it is set. It seems not to be $_GET[‘s’] and not $wp_query->query_vars[‘s’].

Does somebody have an idea where that $s comes from?