Why does my 98 Lexus GS400 not want to turn over after sitting all night?

What is your question?

All dash indicators come on. Radio plays. Headlights work. Windows go up and down. Turn the key and nothing. No engine crank,no noise, no start. Only way I can get it to start is by rocking in my seat, left to right, hard. The movement of the car seems to make a difference. Battery and alternator are new. Please help. My wife wants me to get rid of it but I like it otherwise.

What makes this problem better or worse?

Sitting all night. Will start fine all day after the initial start.

How long has your 1998 Lexus GS400 had this problem?

A few weeks