How to use custom google search on https to avoid non secure content prompt on https

Question

Is there a way to get the script for search results from an HTTPS site instead of http://www.google.com/afsonline/show_afs_search.js

I am using the custom google business search on a HTTPS site. When search is submitted web browser shows warning:

"This page contains both secure and nonsecure items"

I tried to modify the source to be https://www.google.com/afsonline/show_afs_search.js but that doesn't work. Still JavaScript returned from that link connects to http links instead of https links.

Does anybody knows how to fix this ?

Thanks in advance.