Oops!
Your web server is Apache
So, see this from laravel.com:
Apache
The framework ships with a public/.htaccess file that is used to allow URLs without index.php. If you use Apache to serve your Laravel application, be sure to enable the mod_rewrite module.
If the .htaccess file that ships with Laravel does not work with your Apache installation, try this one:
Options +FollowSymLinks
RewriteEngine On
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^ index.php [L]
If your web host doesn't allow the FollowSymlinks option, try replacing it with Options +SymLinksIfOwnerMatch.