I am trying to calculate the z-score for a normal random variable(x), which has a negative value.
I know the formula. If x < 3.4 the mu is 1 and sigma is 16, then Φ is 0.15 which has a probability of .5596[NCST]
How would I calculate x <-3.4?

To use the table with negative quantile, one just need to note that the standard normal distribution is symmetric about 0, i.e. we have

And once you know how to calculate the probability for a complementary event, you can calculate the required probability within the precision of the given table.