Buried in the comments of an androidcentral.com article showing a code to unlock "Service Mode," which allows for the testing of many features on the S3, I found the trick to open the gates of Valhalla! This works on my 4.1.2
The secret is to first type *#22745927 to get to an 'enable/disable hidden menu' screen. Check enable and you're good to go.
The code *#0*# gives this testing screen: 
Also, the code *#0011# works to access "Service Menu" and change the Wifi Power Save to off. It is thought that this might solve the Wifi disconnecting problem for some people.
Here's the link where I found it: How to uncover and use the hidden Service menu on the Galaxy S3 | Android Central
These codes work, too, though I've only tested a couple:
- *#1234# (View SW Version PDA, CSC, MODEM)
- #7465625782*# (Insert partial Network Lock Keycode)
- *746562527*# (Insert Network Lock Keycode NSP/CP)
- #746562527*# (Insert Content Provider Keycode)
- *746562577*# (Insert Network Lock Keycode SP)
- #746562577*# (Insert Operator Lock Keycode)
- #7465625638*# (Insert Network Lock Keycode)
- *#2663# (TSP / TSK firmware update)
- *#7780# (Factory Reset)
- *27673855# (Full Factory Reset)
- *#03# (NAND Flash S/N)
- *#12580369# (SW & HW Info)
- *#197328640# (Service Mode)
- *#0*# (General Test Mode)
- *#0228# (ADC Reading)
- *#7465625# (View Phone Lock Status)
- *7465625638*# (Configure Network Lock MCC/MNC)
- *7465625782*# (Configure Network Lock NSP)
- *#32489# (Ciphering Info)
- *#7284# (USB I2C Mode Control)
- *#272imei#* Product code
- *#9090# (Diagnostic Configuration)
- *#232337# (Bluetooth Address)
- *#232331# (Bluetooth Test Mode)
- *#232338# (WLAN MAC Address)
- *#232339# (WLAN Test Mode)
- *#0842# (Vibra Motor Test Mode)
- *#872564# (USB Logging Control)
- *#4238378# (GCF Configuration)
- *#0283# (Audio Loopback Control)
- *#1575# (GPS Control Menu)
- *#3214789650# (LBS Test Mode)
- *#745# (RIL Dump Menu)
- *#746# (Debug Dump Menu)
- *#9900# (System Dump Mode)
- *#44336# (software Version Info)
- *#0289# (Melody Test Mode)
- *#7594# (Remap Shutdown to End Call TSK)
- *#34971539# (Camera Firmware Update)
- *#8736364# (OTA Update Menu)
- *#526# (WLAN Engineering Mode)
- *#528# (WLAN Engineering Mode)
- *#7412365# (Camera Firmware Menu)
- *#0589# (Light Sensor Test Mode)
- *#273283255663282*# (Data Create SD Card)
- *#3282727336*# (Data Usage Status)
- *#301279# (HSDPA/HSUPA Control Menu)
- *#7353# (Quick Test Menu)
- *#2263# (RF Band Selection)
- *27674387264636# (Sellout SMS / PCODE view)
- *#80# (Unknown)
- *#0782# (Real Time Clock Test)
- *#0673# (Audio Test Mode)
- *#07# (Test History)
- *#3214789# (GCF Mode Status)
- *#272886# (Auto Answer Selection)
- *#0588# (Proximity Sensor Test Mode)
- *#2732832553282*# (Data Create Menu)
By below link
