1935 G
1935 G silver certificates were printed in two varieties:
Despite its date, the "G" series letter indicates your bill was printed in the early 1960s. During that time it was the practice to change a series date only when a bill underwent a major redesign, as opposed to the current practice of changing dates when a new Treasury Secretary is appointed.
Blue Ink
Silver certificates typically had blue seals and serial numbers. Other colors were used on some very old bills and on bills printed for use in threatened regions during WWII.
In God We Trust
That motto was first added to $1 bills starting with the 1957 series and was phased in on other series and denominations over the next few years. The 1935 G series was printed during that transitional time which is why some do and some don't have the motto.
Serial Numbers
A bill's serial number is just a counter so including it in a question generally doesn't help to identify it or affect its value. There are collectors who will pay a premium for bills with low or patterned serial numbers; e.g.12345678 or 00000005.
The serial number is almost never a determining factor in a bill's value. Condition, year, series letter, and bank district are far more important.
There are 2 varieties of this bill. Those without the motto In God We Trust are worth about $2.25-$3.; those with the motto on the back are worth $4-$6 depending on condition.
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Please check your bill again. Its date should be 1935. There's more information at the question "What is the value of a 1935 US 1 dollar bill with HAWAII on it?"
Please take another look at the bill. It's dated 1935, not 1937. There's more information at the question "What is the value of a 1935 US 1 dollar bill with HAWAII on it?"
Please check your bill again. The date should be 1934. The only bills dated 1935 were $1 silver certificates. There's more information at the question "What is the value of a 1934 US 20 dollar bill with HAWAII on it?"
The blue seal indicates your bill is a silver certificate, a form of paper money issued until the early 1960s. Please see the question "What is the value of a 1935 A US 1 dollar silver certificate?" for more information.
Please check the date again - the last two digits are reversed. HAWAII-overprint bills are dated 1935.There's more information at the question "What is the value of a 1935 US 1 dollar bill with HAWAII on it?".