| L(s) = 1 | − 1.73·2-s + 2.73·3-s + 0.999·4-s + 5-s − 4.73·6-s − 2.73·7-s + 1.73·8-s + 4.46·9-s − 1.73·10-s + 4.73·11-s + 2.73·12-s − 4·13-s + 4.73·14-s + 2.73·15-s − 5·16-s − 17-s − 7.73·18-s − 1.46·19-s + 0.999·20-s − 7.46·21-s − 8.19·22-s − 8.19·23-s + 4.73·24-s + 25-s + 6.92·26-s + 3.99·27-s − 2.73·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.57·3-s + 0.499·4-s + 0.447·5-s − 1.93·6-s − 1.03·7-s + 0.612·8-s + 1.48·9-s − 0.547·10-s + 1.42·11-s + 0.788·12-s − 1.10·13-s + 1.26·14-s + 0.705·15-s − 1.25·16-s − 0.242·17-s − 1.82·18-s − 0.335·19-s + 0.223·20-s − 1.62·21-s − 1.74·22-s − 1.70·23-s + 0.965·24-s + 0.200·25-s + 1.35·26-s + 0.769·27-s − 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8213918690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8213918690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 0.535T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−14.18010413167243495194910902705, −13.50753370512687398199216081828, −12.18240140832389069041995272520, −10.22908803495156393026872439186, −9.492184568279700076490858135884, −8.969589372484756977018515056691, −7.79032767204837674777103178108, −6.63915818226960438386848743997, −3.95656870881855527374068847494, −2.17582772087516134866370577695,
2.17582772087516134866370577695, 3.95656870881855527374068847494, 6.63915818226960438386848743997, 7.79032767204837674777103178108, 8.969589372484756977018515056691, 9.492184568279700076490858135884, 10.22908803495156393026872439186, 12.18240140832389069041995272520, 13.50753370512687398199216081828, 14.18010413167243495194910902705
