| L(s) = 1 | − 2-s − 3-s − 2·4-s − 2·5-s + 6-s − 5·7-s + 3·8-s − 4·9-s + 2·10-s + 2·12-s − 5·13-s + 5·14-s + 2·15-s + 16-s − 5·17-s + 4·18-s + 4·20-s + 5·21-s + 23-s − 3·24-s − 7·25-s + 5·26-s + 6·27-s + 10·28-s + 8·29-s − 2·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s − 0.894·5-s + 0.408·6-s − 1.88·7-s + 1.06·8-s − 4/3·9-s + 0.632·10-s + 0.577·12-s − 1.38·13-s + 1.33·14-s + 0.516·15-s + 1/4·16-s − 1.21·17-s + 0.942·18-s + 0.894·20-s + 1.09·21-s + 0.208·23-s − 0.612·24-s − 7/5·25-s + 0.980·26-s + 1.15·27-s + 1.88·28-s + 1.48·29-s − 0.365·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 167 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 31 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 117 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 19 T + 225 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 139 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 293 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−12.25047198128916819428580278737, −12.10344489296036794817082750402, −11.75999058567646841414051267325, −11.00264520804220653887930141660, −10.31459274500394545140246311737, −10.08030900339094430248252807123, −9.407135099814296032923668235565, −9.109231719281949975091417041726, −8.485161563929556055614154271202, −8.167241448518487695616982907009, −7.33234652955298690478844263890, −6.77212364193621050414832681660, −6.19595447725949612902325342091, −5.57068769982988803465290620174, −4.71806617982442292753682333899, −4.28237281363890767911049859818, −3.29035960345559828858033705025, −2.68748749398594051914538701024, 0, 0,
2.68748749398594051914538701024, 3.29035960345559828858033705025, 4.28237281363890767911049859818, 4.71806617982442292753682333899, 5.57068769982988803465290620174, 6.19595447725949612902325342091, 6.77212364193621050414832681660, 7.33234652955298690478844263890, 8.167241448518487695616982907009, 8.485161563929556055614154271202, 9.109231719281949975091417041726, 9.407135099814296032923668235565, 10.08030900339094430248252807123, 10.31459274500394545140246311737, 11.00264520804220653887930141660, 11.75999058567646841414051267325, 12.10344489296036794817082750402, 12.25047198128916819428580278737
