| L(s) = 1 | + (0.106 − 0.00478i)2-s + (−1.98 + 0.178i)4-s + (−0.619 − 0.201i)5-s + (−0.0215 − 0.0780i)7-s + (−0.421 + 0.0570i)8-s + (−0.0670 − 0.0184i)10-s + (1.23 + 2.28i)11-s + (−1.92 − 1.03i)13-s + (−0.00266 − 0.00820i)14-s + (3.86 − 0.702i)16-s + (5.32 − 3.86i)17-s + (2.21 − 5.19i)19-s + (1.26 + 0.288i)20-s + (0.141 + 0.237i)22-s + (3.47 − 4.36i)23-s + ⋯ |
| L(s) = 1 | + (0.0753 − 0.00338i)2-s + (−0.990 + 0.0891i)4-s + (−0.277 − 0.0900i)5-s + (−0.00813 − 0.0294i)7-s + (−0.149 + 0.0201i)8-s + (−0.0211 − 0.00584i)10-s + (0.370 + 0.689i)11-s + (−0.533 − 0.287i)13-s + (−0.000712 − 0.00219i)14-s + (0.967 − 0.175i)16-s + (1.29 − 0.937i)17-s + (0.509 − 1.19i)19-s + (0.282 + 0.0645i)20-s + (0.0302 + 0.0506i)22-s + (0.725 − 0.909i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921938 - 0.495771i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921938 - 0.495771i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (7.68 + 3.45i)T \) |
| good | 2 | \( 1 + (-0.106 + 0.00478i)T + (1.99 - 0.179i)T^{2} \) |
| 5 | \( 1 + (0.619 + 0.201i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.0215 + 0.0780i)T + (-6.00 + 3.59i)T^{2} \) |
| 11 | \( 1 + (-1.23 - 2.28i)T + (-6.05 + 9.18i)T^{2} \) |
| 13 | \( 1 + (1.92 + 1.03i)T + (7.16 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-5.32 + 3.86i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.21 + 5.19i)T + (-13.1 - 13.7i)T^{2} \) |
| 23 | \( 1 + (-3.47 + 4.36i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (4.65 - 7.79i)T + (-13.7 - 25.5i)T^{2} \) |
| 31 | \( 1 + (-1.90 + 10.5i)T + (-29.0 - 10.8i)T^{2} \) |
| 37 | \( 1 + (1.73 + 2.18i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (-6.96 + 3.35i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (6.71 - 2.51i)T + (32.3 - 28.2i)T^{2} \) |
| 47 | \( 1 + (-6.19 - 5.41i)T + (6.30 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-8.21 - 0.738i)T + (52.1 + 9.46i)T^{2} \) |
| 59 | \( 1 + (0.998 - 1.04i)T + (-2.64 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 5.15i)T + (-52.3 - 31.2i)T^{2} \) |
| 67 | \( 1 + (0.365 + 4.06i)T + (-65.9 + 11.9i)T^{2} \) |
| 73 | \( 1 + (-0.592 - 13.2i)T + (-72.7 + 6.54i)T^{2} \) |
| 79 | \( 1 + (-1.39 - 10.2i)T + (-76.1 + 21.0i)T^{2} \) |
| 83 | \( 1 + (4.96 + 4.74i)T + (3.72 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-1.14 + 12.7i)T + (-87.5 - 15.8i)T^{2} \) |
| 97 | \( 1 + (-7.16 + 14.8i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.21786107736736609751075463444, −9.497741398209840078466180812984, −8.886144409022585282897656689583, −7.71481496127253032853069230398, −7.14910039428160754008567785787, −5.63539174891144538823410481100, −4.85566842894629745593752686057, −3.96325750623193975105644743566, −2.72653274808565603550919578923, −0.67263096248985601851253534738,
1.30514826180504152212825100613, 3.36349100392200595769248737357, 3.99126624419432671610883937299, 5.34892975589745516608195419691, 5.91771988619442135397201690458, 7.38618158850120105239726025704, 8.121207504817826757131958312588, 9.005757295897484008945576887324, 9.832700067038682605848313095914, 10.50732691383174202212384642346
