1. A system for providing low-latency computational capacity in an on-demand code execution environment, the system comprising: a data store including historical data regarding use of the on-demand code execution environment, the historical data correlating pre-trigger notifications previously obtained at the on-demand code execution environment with one or more requests to execute first user-specified code obtained at the on-demand code execution environment subsequent to individual pre-trigger notifications, wherein the pre-trigger notifications are obtained from one or more computing systems associated with the user-specified code and distinct from the on-demand code execution environment, and wherein the pre-trigger notifications are requests to execute second user-specified code that are distinct from the requests to execute the first user-specified code that are obtained subsequent to the individual pre-trigger notifications; and one or more computing devices in communication with the data store and executing the computer-executable instructions, wherein execution of the computer-executable instructions causes the one or more computing devices to: maintain a set of virtual machine instances that are pre-initialized with an operating environment and made available for execution of the first user-specified code; obtain, from the one or more computing systems, an additional pre-trigger notification requesting execution of the second user-specified code; determine, based at least in part on comparing the additional pre-trigger notification to the historical data correlating pre-trigger notifications previously obtained at the on-demand code execution environment, a number of requests to execute the first user-specified code that are predicted to occur subsequent to obtaining the additional pre-trigger notification; based on the number of requests to execute the first user-specified code that are predicted to occur subsequent to obtaining the additional pre-trigger notification, increase a number of virtual machine instances within the set of virtual machine instances at least partly by initializing at least one new virtual machine instance with an operating environment and adding the at least one new virtual machine instance to the set of virtual machine instances; obtain a request to execute the first user-specified code, wherein the request is associated with the pre-trigger notification; and execute the first user-specified code on the previously initialized operating environment of the at least one new virtual machine instance.