The problem you are experiencing is due to low light levels. You can see the seedlings are tending to bend over in one direction (presumably towards where the light source was). UV won't solve your problems - much of the light plants need is in the visible spectrum. A grow light that has actually been developed for plant growth should improve things. In this instance, the temperature you mention won't have much of an impact. Higher temperatures will increase the metabolic rate of a plant meaning that they will need to photosynthesis more rapidly to keep up with their energy requirements (and therefore need brighter light).
Another issue is that these are massively overcrowded. Coriander (Coriandrum sativum) can grow up to a 1m tall in my garden (UK). This is going to cause issues - firstly, it increases competition for light. Secondly, at some point, one of those seedlings is going to die and with the rest packed so tightly the others around it will die as well. That pot is only big enough for one plant, so put two or three seeds in to start and thin out to one at the first true leaf stage. You may see pots with lots of seedlings like this being sold in the supermarket - these are produced in ideal conditions and are not intended for long term growth. They are really just a way of selling a bunch of coriander that will stay fresh for longer.