Call product search from custom hook

I am trying to make search request from my custom form. For this I have an AJAX call on form submit:

jQuery("#header-search").on('submit', function () {
        jQuery.ajax({
            url: my_ajax_object.ajax_url,
            data: jQuery(this).serialize()
        })
        .then(function (res) {
            console.log(res)
        })
        return false
    })

In my functions.php I have:

function custom_search () {
    //WHAT SHOULD I CALL HERE
}
add_action( 'wp_ajax_nopriv_custom_search', 'custom_search' );
add_action( 'wp_ajax_custom_search', 'custom_search' );

I am not sure if this should be made as AJAX but I don’t really understand what action should I put on the search form in the HTML. Everything seems like a mess for me. Can you assist me this? Appreciating every suggestion.