| L(s) = 1 | + 4-s − 2·7-s − 3·9-s − 3·16-s − 10·25-s − 2·28-s − 3·36-s − 4·37-s − 16·43-s − 3·49-s + 6·63-s − 7·64-s + 28·67-s − 8·79-s + 9·81-s − 10·100-s + 20·109-s + 6·112-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 9·144-s − 4·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 9-s − 3/4·16-s − 2·25-s − 0.377·28-s − 1/2·36-s − 0.657·37-s − 2.43·43-s − 3/7·49-s + 0.755·63-s − 7/8·64-s + 3.42·67-s − 0.900·79-s + 81-s − 100-s + 1.91·109-s + 0.566·112-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/4·144-s − 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.604493874916442322618985415366, −9.102465290136124237335587161787, −8.403975666931309373366250948631, −8.191121791582851434996242624931, −7.46392714285653844602538121483, −6.86014425374360804755831712438, −6.45351506973656977615701546788, −5.98619691132085661379852739932, −5.35040149895477903209300448936, −4.80113720358060376916394664785, −3.78634887396349313968134791064, −3.40535175182779373850407284128, −2.53997112478931449817035822477, −1.87611291879719114111766464275, 0,
1.87611291879719114111766464275, 2.53997112478931449817035822477, 3.40535175182779373850407284128, 3.78634887396349313968134791064, 4.80113720358060376916394664785, 5.35040149895477903209300448936, 5.98619691132085661379852739932, 6.45351506973656977615701546788, 6.86014425374360804755831712438, 7.46392714285653844602538121483, 8.191121791582851434996242624931, 8.403975666931309373366250948631, 9.102465290136124237335587161787, 9.604493874916442322618985415366
