| L(s) = 1 | + 0.561·2-s − 7.68·4-s − 6.68·5-s + 7·7-s − 8.80·8-s − 3.75·10-s + 11·11-s + 14.3·13-s + 3.93·14-s + 56.5·16-s + 47.7·17-s + 11.9·19-s + 51.3·20-s + 6.17·22-s + 44.4·23-s − 80.3·25-s + 8.03·26-s − 53.7·28-s + 139.·29-s − 208.·31-s + 102.·32-s + 26.8·34-s − 46.7·35-s − 253.·37-s + 6.69·38-s + 58.8·40-s + 156.·41-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.960·4-s − 0.597·5-s + 0.377·7-s − 0.389·8-s − 0.118·10-s + 0.301·11-s + 0.305·13-s + 0.0750·14-s + 0.883·16-s + 0.681·17-s + 0.143·19-s + 0.574·20-s + 0.0598·22-s + 0.403·23-s − 0.642·25-s + 0.0605·26-s − 0.363·28-s + 0.893·29-s − 1.20·31-s + 0.564·32-s + 0.135·34-s − 0.225·35-s − 1.12·37-s + 0.0285·38-s + 0.232·40-s + 0.595·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 0.561T + 8T^{2} \) |
| 5 | \( 1 + 6.68T + 125T^{2} \) |
| 13 | \( 1 - 14.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 36.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 114.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 53.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 132.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 817.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 369.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 69.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 467.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−9.579425032450930458840847296578, −8.685274798969743728111074218185, −8.053926276143849475536081811005, −7.09458888031229806519488101139, −5.82932443745915256131683548196, −4.97061842491356212840182660399, −4.03273550061096967927586777432, −3.21851181291526248165313877429, −1.37622371514551312910643013372, 0,
1.37622371514551312910643013372, 3.21851181291526248165313877429, 4.03273550061096967927586777432, 4.97061842491356212840182660399, 5.82932443745915256131683548196, 7.09458888031229806519488101139, 8.053926276143849475536081811005, 8.685274798969743728111074218185, 9.579425032450930458840847296578
