The probability at least one number doesn't show up is a bit over 56%. I did this by random simulations of die rolling rather than working out an exact probability because it was too tedious.
For fun, I wrote a script that handles variations of this problem. You can change the number of times you roll the die and the number of sides of the die to estimate the likelihood at least one number doesn't show up.
https://sagecell.sagemath.org/?z=eJx1UNFOhDAQfCfhHyYYY3uHHtyDDyb8g-_GXAoUrSkt6e7F3N_bIidi4r603c7szGzHaFAhz8iMVCJ4a-PRG31y5_FEptcUAccqVYn6WOIRN4h_rQ7wA0iNk9V0iPSzVWy8i_T1e5nHl0mnZ5ybZ4MPMDAOQbk3LZKwfMozxEpmXoxjsTGwi8Dej0JKJO7Hyp3Hy9dvshlgtROkWXCENs02xiKRqmPsG9RrI8-mkGSL5-Bb1Rpr-AJlLQrsQRy2fmRsFktIAr37z2j7ihysVyzqqtp1fJizzfDbh0L-qPy6bgQ5BlDE8E5fd9h7Te6O_1eZ93ffsfwj9gU2q5BI&lang=python&interacts=eJyLjgUAARUAuQ==
For example, if you keep rolls = 12 but change the number of sides from 6 to 8 (octahedral die), the probability not all numbers show up is between 90% and 91%.
Or, if you keep the number of sides = 6 but change the number of rolls from 12 to 13, the probability not all numbers show up is now between 48% and 49%.
I have the number of samples/simulations set to 200000 because that's toward the upper end of what the sagecell server can handle. The server will often timeout if you set it to a million, in my experience. If you think of other ways to vary the problem I can update the script to include extra variables.