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If you search empty with wordpress such as url shows: ?s=&
Example: http://page60.com/?s=&searchsubmit.x=0&searchsubmit.y=0
Then template hierarchy page base will be index.php in theme hierarichy

If you search with some word/char such as url shows: ?s=abc
Example: http://page60.com/?s=abc&searchsubmit.x=0&searchsubmit.y=0
then first priority will be of search.php as a base then index.php