|"The thing is that I can connect to all servers with any desktop computer in the network, but when I try to do it in my laptop (that is connected to the same LAN) I've got access to the other servers but not to the web server"|
Whatever is happening seems to be happening on your laptop. At least, what you say above points directly at the laptop as the culprit:
As to what could be happening on the laptop to cause it not to connect to the web server is something that would require investigation and troubleshooting, none of which can really be done from here. If it were me, I'd check all TCP/IP settings on the laptop, I'd check the NIC (drivers, update them if there are newer). I would look at the TCP/IP stack and possibly uninstsall/reinstall it.
If still no joy, you may need to go so far as to do a complete, clean, reinstall of the OS (operating system) on the laptop. If that doesn't do the trick you'll need to replace that laptop with one that doesn't share this problem.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.