Phoned customer support. After an hour disconnecting the modem they asked to check for http://22.214.171.124 . Because I could see this webpage, they said the automatic DNS server does not give to my router whatever it is.
Instead I needed to input manually into DNS-servers on the router (or under LAN-connection, TCP/IP on the computer):
I actually had a similar problem after updating the firmware of my modem recently. It seemed to get the right address but did not assign the right address to my router/computer so i had to set it manually like you did.