Are you wearing gloves? Do they have padding?

There are many nerve endings in the hand and it's very common for them to go numb depending on how and where you're gripping the bars. If you get gloves with even a small amount of padding, it can help greatly.

Try shifting your grip around, sometimes the slightest correction will do the trick. Even when standing, you shouldn't be gripping the bars so hard that you can't wiggle your fingers.