Here is a screenshot:
I was very excited to get a hand-down iPhone 5s, then now I am very upset to find it kaput. I believe prfc: it makes sense that this error, "The iPhone" "could not be restored. An unknown error occurred (40).", indicates that flash memory is detaching / separating from mainboard. Boot code in ROM initiates backlight, screen is fully blank black. iTunes can attempt to install operating system to flash memory, is not successful (error 40) because memory package is becoming disconnected. How disheartening.
As of 2017-02-07, Apple (United States) repair price of iPhone 5s is $ 269. I am not capable of reflowing anything nearly as high-tech as iPhone. (I am a beginner solderer.) Maybe after next new iPhone is released, price for pre-used logic boards will be lower, and maybe iFixit will have them in stock.
Addition 2017-02-12: I thought "Not much to lose, go ahead and toast it!", spent last few hours trying to reflow my 5S logic board. I did not do it "the right way" by removing shield. I just blew it with hot air, shield still on, over where NAND package can be found. This was exercise was futile: iPhone still is useless, iTunes still results error 40. Here is a snapshot of iPhone screen, with its empty progress bar: