Well, this is a privilege. Five years ago I was peripherally involved with the restoration of Jaguar's XJ13, the one-off Le Mans hope that time overtook. This mid-engined, louvred-and-riveted machine with its 5.0-litre, 502bhp V12, the face of a current XK and the tail-lights of an E-type, is surely one of the best-looking racing cars ever made. Best-sounding, too. And now I'm about to blast it up the hill at the Goodwood Festival of Speed.

All eyes are on me as I sink down into the cosy cockpit, the one in whose footwell ace Jaguar test driver Norman Dewis cowered as the XJ13 famously performed its 1971 barrel-roll at the MIRA test track during filming for the then-new XJ12 saloon's launch film.

All eyes, that is, except the vast numbers on comedian and talk-show host Jay Leno who is firing up the scary monster next to me, the ex-Bob Tullius Group 44 E-type V12. There are a lot of V12 decibels in the paddock right now.

The XJ13 fits me perfectly because most 1960s racing drivers were not tall. Fuel pump on, ignition on, spring the ignition-toggle down to trigger the starter, hear/feel the eruption aft.

First gear in the semi-sequential ZF gearbox can't be found without selecting third, then second, beforehand; now we're in gear and trickling through the paddock, engine idling with surprising docility but clearly keen to clear its throat. There are no seatbelts, and the view through the Perspex windscreen is worryingly ripply.

Down at the startline among a throng of Le Mans and other sports-racing cars through the ages – new Audis, old Porsches, the 1991-winning Mazda 787B of ear-bending sonic ferocity, the first McLaren M1B CanAm car, Jaguars of all flavours – Leno has just lit up the E-type's rear tyres and bellowed off in a cloud of smoke. My turn next.

Quick practice start on the tyre-warming strip: traction breaks satisfyingly, XJ13 snakes and surges forward. Now for the real thing. About 3500rpm, drop the clutch, feed in the power as the XJ13 bellows like the E-type but with a more manic, metallic edge. Great start, though I say so myself. Second gear, third, blip-shift back to second for the first corner, back to third in front of Goodwood House and Gerry Judah's giant E-type celebratory sculpture made from lengths of large-bore pipe.

Now the tricky twist past the wall, second gear, then back to third and power through to the steepening section of the hill. This is fabulous; crowds are cheering, the engine is howling, now I'm in fourth and hurtling towards the finish, exercising the XJ13 as it was meant to be exercised. I'm using full throttle, but it's not scary because the steering and balance are sublime.

Actually, the pace is not quite as ferocious as it was the first time I drove the XJ13 in anger, because its custodians at Jaguar Heritage have backed off the ignition advance to make the engine last longer. Fair enough; this is the only complete XJ13 four-cam engine they possess.

On my return to the paddock, someone tells me the XJ13 is the pick of the event. This is a car lodged in legend, even though it never raced because by the time it was ready, the rules for the prototype class had changed to a 3.0-litre limit. And even if it had been ready for the 1967 season as originally hoped, it couldn't have conquered the Ford GT40 MkIVs and Ferrari P4s. It had to remain a dream… and I have just lived a part of that dream.

Text: John Simister
Photos: Jaguar