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The seven-time world champion missed a Pirelli tyre test in Italy as well as an event at Milan Fashion Week last week after the 12-year-old dog caught pneumonia and was placed in a coma. He said Roscoe was put to sleep on Sunday, in his arms.
Hamilton has never gone through a complete season without a podium finish and has faced mounting criticism of his declining form, including a warning from former F1 ringmaster Bernie Ecclestone to retire before he is forgotten.
Qualifying takes place from around 6 PM Kenyan time live from the Red Bull Ring in Salzburg.
The seven-time world champion asked reporters to "stop making stuff up" as he delivered an impassioned response to the reports on the eve of this weekend's Canadian Grand Prix, an event he has won a record-equalling seven times.
Earlier this week, Germany's Bild newspaper reported that Ferrari had recently sounded out the Briton as a possible replacement for Fred Vasseur.
Hamilton has yet to win a place on the podium this season and has struggled to get his car going, trailing in the wake of McLaren's upwards trajectory and his learning curve with a new vehicle in his hands.
Norris' victory cut Piastri's lead at the head of the championship to three points, with Verstappen a further 22 behind in third.
The four-time champion passed pole-sitter Oscar Piastri's McLaren around the outside of the first chicane.
Piastri edged Verstappen by 0.034 seconds despite encountering traffic in the final two corners.
Less than a 10th of a second covered the top five cars, as Williams driver Carlos Sainz took third place ahead of Mercedes' George Russell and Ferrari's Lewis Hamilton.
Piastri won the Miami Grand Prix on Sunday, as McLaren secured a dominant one-two finish.
The 18-year-old beat McLaren's Oscar Piastri, the championship leader, by 0.045 seconds.