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GE gas range 14 years old, model jgbp30bew1ad. No heat on bake, no
heat on broil, stove works fine.

Is it likely the gas valve burnt out, on both bake and broil, due to
this shorting out of the bake igniter? Or maybe it's the control
board? I still hear a relay click on when I turn the oven knob on for
bake, same for broil. And even self-clean cycle indicators light up
and it starts a count-down, of course there's no heat in the
oven...Maybe just the thermostat? Or is it just time to get a new
range...

Thanks.

Most likely a safety/sensor of some sort that will not allow the valve to
open. At 14 years, you may want to consider replacing rather than put a
couple of hundred into diagnostics and repair. Sure, it can last another
five or ten years, but just don't dump a tone of $$$.

You may want to check out www.repairclinic.com for some diagnostics.