Is it possible to set Cache-Control / Expires headers for 404 responses in Nginx?

I've tried using an error_page like this:

[code]
location ^~ /x/ {
error_page 404 /not_found.html;
}
location /not_found.html {
internal;

root /static/errordoc;
expires 10m;
add_header Cache-Control "private, max-age=600";
}
[/code]

But no luck.

Thanks in advance!