I know this one didn't work for NYC1, however maybe someone in Europe could try this number:

+41 848 365 365

I did call from the US, and when they answered and I asked, they told me that it was indeed the LX first class number, and I was successful in reaching an agent there. YMMV, but it's worth giving it a try to see if it's a good number, and one you may want to keep on hand. I'll also note that this was the number given to me by the LH first class line desk in Germany. Unlike the LH number, it is NOT toll free and is not a free Skype call.