| L(s) = 1 | − 0.741·2-s − 1.44·4-s + 3.30·5-s + 4.44·7-s + 2.55·8-s − 2.44·10-s − 4.04·11-s + 2·13-s − 3.30·14-s + 1.00·16-s − 6.60·17-s + 19-s − 4.78·20-s + 3·22-s + 4.78·23-s + 5.89·25-s − 1.48·26-s − 6.44·28-s + 2.55·29-s + 7.44·31-s − 5.86·32-s + 4.89·34-s + 14.6·35-s + 4.44·37-s − 0.741·38-s + 8.44·40-s + 3.70·41-s + ⋯ |
| L(s) = 1 | − 0.524·2-s − 0.724·4-s + 1.47·5-s + 1.68·7-s + 0.904·8-s − 0.774·10-s − 1.21·11-s + 0.554·13-s − 0.882·14-s + 0.250·16-s − 1.60·17-s + 0.229·19-s − 1.07·20-s + 0.639·22-s + 0.997·23-s + 1.17·25-s − 0.291·26-s − 1.21·28-s + 0.475·29-s + 1.33·31-s − 1.03·32-s + 0.840·34-s + 2.48·35-s + 0.731·37-s − 0.120·38-s + 1.33·40-s + 0.579·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.360511045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360511045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.741T + 2T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 0.741T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 8.34T + 61T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 5.86T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 5.55T + 97T^{2} \) |
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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
−10.69800532021932363536123007533, −10.03575507662976454533541442618, −8.987933457460651954525313612403, −8.448888484012085320515324226519, −7.55702636362463005890096604035, −6.17551082777687147027203478699, −5.04844163673424035729343173221, −4.62778454936819792621912895530, −2.47066633600105172875040219039, −1.33459346389092772668776637293,
1.33459346389092772668776637293, 2.47066633600105172875040219039, 4.62778454936819792621912895530, 5.04844163673424035729343173221, 6.17551082777687147027203478699, 7.55702636362463005890096604035, 8.448888484012085320515324226519, 8.987933457460651954525313612403, 10.03575507662976454533541442618, 10.69800532021932363536123007533
