| L(s) = 1 | − 1.32·2-s − 3·3-s − 6.23·4-s − 15.4·5-s + 3.98·6-s + 7.96·7-s + 18.9·8-s + 9·9-s + 20.4·10-s − 12.7·11-s + 18.7·12-s − 10.5·14-s + 46.2·15-s + 24.8·16-s − 54·17-s − 11.9·18-s + 84.5·19-s + 96.2·20-s − 23.8·21-s + 16.9·22-s + 122.·23-s − 56.7·24-s + 112.·25-s − 27·27-s − 49.6·28-s + 140.·29-s − 61.4·30-s + ⋯ |
| L(s) = 1 | − 0.469·2-s − 0.577·3-s − 0.779·4-s − 1.37·5-s + 0.270·6-s + 0.430·7-s + 0.835·8-s + 0.333·9-s + 0.647·10-s − 0.350·11-s + 0.450·12-s − 0.201·14-s + 0.796·15-s + 0.387·16-s − 0.770·17-s − 0.156·18-s + 1.02·19-s + 1.07·20-s − 0.248·21-s + 0.164·22-s + 1.11·23-s − 0.482·24-s + 0.903·25-s − 0.192·27-s − 0.335·28-s + 0.901·29-s − 0.373·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.32T + 8T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 - 7.96T + 343T^{2} \) |
| 11 | \( 1 + 12.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 433.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 485.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 674.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 186.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 671.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 14.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 346.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 832.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 568.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 236.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−10.04212159467590917532284739884, −9.135001840842466875617976315188, −8.066517863177622393090774259296, −7.70353150669798401134992613335, −6.49564873042756857238405258413, −4.91429409181560840500400367536, −4.55569368602154890799671881204, −3.24057348303170224625921071912, −1.09740346971473203087223673547, 0,
1.09740346971473203087223673547, 3.24057348303170224625921071912, 4.55569368602154890799671881204, 4.91429409181560840500400367536, 6.49564873042756857238405258413, 7.70353150669798401134992613335, 8.066517863177622393090774259296, 9.135001840842466875617976315188, 10.04212159467590917532284739884
