| L(s) = 1 | − 4.54·2-s + 12.6·4-s − 13.1·5-s + 7·7-s − 21.0·8-s + 59.7·10-s + 11·11-s − 54.8·13-s − 31.7·14-s − 5.62·16-s + 79.9·17-s + 21.2·19-s − 166.·20-s − 49.9·22-s − 119.·23-s + 48.3·25-s + 249.·26-s + 88.3·28-s + 87.4·29-s + 191.·31-s + 193.·32-s − 362.·34-s − 92.1·35-s + 91.6·37-s − 96.5·38-s + 276.·40-s + 60.4·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.57·4-s − 1.17·5-s + 0.377·7-s − 0.928·8-s + 1.89·10-s + 0.301·11-s − 1.16·13-s − 0.606·14-s − 0.0878·16-s + 1.14·17-s + 0.256·19-s − 1.85·20-s − 0.484·22-s − 1.08·23-s + 0.386·25-s + 1.87·26-s + 0.596·28-s + 0.560·29-s + 1.10·31-s + 1.06·32-s − 1.83·34-s − 0.445·35-s + 0.407·37-s − 0.412·38-s + 1.09·40-s + 0.230·41-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 | \( 1 + 4.54T + 8T^{2} \) |
| 5 | \( 1 + 13.1T + 125T^{2} \) |
| 13 | \( 1 + 54.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 60.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 417.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 414.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 358.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 711.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 131.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 10.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 618.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 534.T + 9.12e5T^{2} \) |
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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
−9.850275722600019857445586262582, −8.573179794888514900965247735255, −7.977559958490869370132602264311, −7.48003209521311523840155229203, −6.54277824958488414116181866851, −5.04500182364427914562724302962, −3.89409295726334963251950373406, −2.50348578530698587819550462608, −1.10613199601176443737102633840, 0,
1.10613199601176443737102633840, 2.50348578530698587819550462608, 3.89409295726334963251950373406, 5.04500182364427914562724302962, 6.54277824958488414116181866851, 7.48003209521311523840155229203, 7.977559958490869370132602264311, 8.573179794888514900965247735255, 9.850275722600019857445586262582
