If you've got a DHCP server, it will most likely also have a DNS server - that would route to the outside.
Check your DNS server - on Windows, using "ipconfig /all" and looking for "DNS Servers:", or on Linux, using "nslookup google.com" and check the "Server:" line (first one of output).
1. Is this a local address, or an address on the Internet?
Furthermore, "doesn't work" is an error description any developer hates
2. Does "doesn't work" mean it doesn't find nslookup?
3. Or that it returns an error code?
4. Or that it doesn't know 1.2.3.4 (You need to use an existing real IP address of one of your servers instead of the 1.2.3.4)?
5. Does "doesn't work" mean that ping does not return an IP address?
6. If so, which name did you supply?
7. Why don't you think that your DNS would resolve local network names? Your knowledge of I networks seems to be limited, so please don't guess
You could also try "hostname -a" on a Linux server to find out it's name.