| L(s) = 1 | − 2-s − 2·3-s + 4-s + 4·5-s + 2·6-s − 2·7-s − 8-s − 4·10-s − 6·11-s − 2·12-s + 2·14-s − 8·15-s + 16-s + 4·20-s + 4·21-s + 6·22-s + 2·23-s + 2·24-s + 4·25-s + 2·27-s − 2·28-s − 8·29-s + 8·30-s + 6·31-s − 32-s + 12·33-s − 8·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 1.78·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s − 1.26·10-s − 1.80·11-s − 0.577·12-s + 0.534·14-s − 2.06·15-s + 1/4·16-s + 0.894·20-s + 0.872·21-s + 1.27·22-s + 0.417·23-s + 0.408·24-s + 4/5·25-s + 0.384·27-s − 0.377·28-s − 1.48·29-s + 1.46·30-s + 1.07·31-s − 0.176·32-s + 2.08·33-s − 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−15.3480846210, −15.0707858435, −14.3128871475, −13.7434342289, −13.2787020247, −13.0440271110, −12.6698479425, −11.8856960747, −11.3601200065, −11.0802456323, −10.3797410601, −10.0501112793, −9.79779537453, −9.27673942192, −8.61032167106, −7.94862413886, −7.43832866615, −6.58940887290, −6.15400690918, −5.85430316267, −5.25152527596, −4.83736845617, −3.30873696736, −2.60744257047, −1.76780396333, 0,
1.76780396333, 2.60744257047, 3.30873696736, 4.83736845617, 5.25152527596, 5.85430316267, 6.15400690918, 6.58940887290, 7.43832866615, 7.94862413886, 8.61032167106, 9.27673942192, 9.79779537453, 10.0501112793, 10.3797410601, 11.0802456323, 11.3601200065, 11.8856960747, 12.6698479425, 13.0440271110, 13.2787020247, 13.7434342289, 14.3128871475, 15.0707858435, 15.3480846210
