http://www.howtojb.com/

From Where you can access iOS 6.1.3 Jailbreak then you have to read furthur more about it.
Apple iOS 6.1.3 Apps and Jailbreak Official Website : http://www.howtojb.com/

Apple Software Update iOS 6.1.3 Compatibility:
iPhone 5
iPhone 4S
iPhone 3GS
iPhone 4GSM Model
iPod Touch 3rd generation
iPod Touch 4th generation
iPad
iPad 2
iPad 3
The following steps will help you jailbreak your device on Apple iOS 6.1.3
Step 1: Go to apple.com and Download latest iTune v 10.2.1 and Update your device to Software Update 6.1.3
Step 2: Then go to http://www.howtojb.com/ and Download iOS 6.1.3 Jailbreak for Mac or Windows
Step 3: Save the iOS 6.1.3 Jailbreak Tool on desktop and Run as Administration
Step 4: Select your device and click on jailbreak button.
Step 5: Let the application do its work and wait for the app to show "Jailbreak Complete!" message.
Step 6.1.3: At this point, wait for the device to automatically reboot otherwise Reboot it according to the messesge.
Step 7: Finish! You should now have a fully jailbroken iPhone & iPad & iPad firmware iOS 6.1.3
[Additional Tags]
Jailbreak 6.1.3
Jailbreak iOS 6.1.3
Jailbreak Firmware 6.1.3
Jailbreak Final iOS 6.1.3
How to Jailbreak 6.1.3
How to Jailbreak iOS 6.1.3
How to Jailbreak firmware 6.1.3
Jailbreak 6.1.3 iPhone
Jailbreak 6.1.3 iphone 4
Jailbreak 6.1.3 iphone 4GSM
Jailbreak 6.1.3 iPhone 3GS
Jailbreak 6.1.3 iPhone 3G
Jailbreak 6.1.3 iPod Touch
Jailbreak 6.1.3 iPod Touch 4