add the below code inside your functions.php, I assume your theme header.php has do_action('wp_head') action filters.
function my_custom_blog_title(){
$blogname = $output = get_option('blogname');
$separator = ' » '; // raquo >> chars
if (is_home()):
$output .= wp_title($separator,false,'left');
elseif (is_search()):
$output = sprintf(__('%1$s - Search results for: %2$s'),$blogname, wp_specialchars(get_query_var('s'), 1));
elseif(is_404()):
$output .= __(' 404 Error: Page not found');
else:
$output = wp_title($separator,false,'right').$blogname;
endif;
$output = '<title>'.$output.'</title>'.PHP_EOL;
echo apply_filters('my_custom_blog_title', $output);
}
add_action('wp_head','my_custom_blog_title');