| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 3·7-s + 4·10-s + 6·14-s − 4·16-s − 6·17-s − 19-s − 4·20-s − 6·23-s − 25-s − 6·28-s + 6·29-s − 7·31-s + 8·32-s + 12·34-s + 6·35-s + 6·37-s + 2·38-s + 8·41-s − 5·43-s + 12·46-s + 2·47-s + 2·49-s + 2·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 1.13·7-s + 1.26·10-s + 1.60·14-s − 16-s − 1.45·17-s − 0.229·19-s − 0.894·20-s − 1.25·23-s − 1/5·25-s − 1.13·28-s + 1.11·29-s − 1.25·31-s + 1.41·32-s + 2.05·34-s + 1.01·35-s + 0.986·37-s + 0.324·38-s + 1.24·41-s − 0.762·43-s + 1.76·46-s + 0.291·47-s + 2/7·49-s + 0.282·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28899 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.84044077760343, −15.66065155212724, −14.79938222136755, −14.20337256665476, −13.44116507873517, −13.03209873220026, −12.45461090905835, −11.79917553442349, −11.22090459168271, −10.86943594066227, −10.18181402563864, −9.597561526863061, −9.364114245574732, −8.490914080622434, −8.280456921409257, −7.598580353318314, −7.004718279322929, −6.534660377546502, −5.978965459180375, −4.953115782387512, −4.062092385542207, −3.885891234413616, −2.717053216633078, −2.216138633818061, −1.170687269301073, 0, 0,
1.170687269301073, 2.216138633818061, 2.717053216633078, 3.885891234413616, 4.062092385542207, 4.953115782387512, 5.978965459180375, 6.534660377546502, 7.004718279322929, 7.598580353318314, 8.280456921409257, 8.490914080622434, 9.364114245574732, 9.597561526863061, 10.18181402563864, 10.86943594066227, 11.22090459168271, 11.79917553442349, 12.45461090905835, 13.03209873220026, 13.44116507873517, 14.20337256665476, 14.79938222136755, 15.66065155212724, 15.84044077760343
