https://www.tech-recipes.com/blog/2006/01/25/google-adsense-sitesearch-have-the-search-box-default-to-local-site/
When using Google SiteSearch, you can have the search box default to Google or to your own site. It is not obvious how to change this choice. Read on to find...
google adsensesearch boxsitesearchdefaultlocal