The search bar in a header will be used for navigation to the search page.
It is a common practice to place search bar on the search page. The user not supposed to scan the page and use header or other type of navigation when they do not need to navigate away from the page
You would want to place search bar to 404 page in order to avoid clicking on “back” button. If you can add header it would be even better. In this case your client will be able to navigate inside your web site and find your content without losing an interest to your site.