| L(s) = 1 | + (0.403 + 1.35i)2-s + (−1.52 + 0.819i)3-s + (−1.67 + 1.09i)4-s + (1.81 − 3.60i)5-s + (−1.72 − 1.73i)6-s + (3.06 − 0.917i)7-s + (−2.15 − 1.82i)8-s + (1.65 − 2.50i)9-s + (5.62 + 1.00i)10-s + (0.0562 − 0.966i)11-s + (1.65 − 3.04i)12-s + (1.00 − 1.35i)13-s + (2.47 + 3.78i)14-s + (0.193 + 6.98i)15-s + (1.60 − 3.66i)16-s + (1.25 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.285 + 0.958i)2-s + (−0.880 + 0.473i)3-s + (−0.837 + 0.546i)4-s + (0.810 − 1.61i)5-s + (−0.704 − 0.709i)6-s + (1.15 − 0.346i)7-s + (−0.762 − 0.646i)8-s + (0.551 − 0.834i)9-s + (1.77 + 0.316i)10-s + (0.0169 − 0.291i)11-s + (0.478 − 0.877i)12-s + (0.279 − 0.375i)13-s + (0.662 + 1.01i)14-s + (0.0500 + 1.80i)15-s + (0.402 − 0.915i)16-s + (0.305 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27755 + 0.283810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27755 + 0.283810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.403 - 1.35i)T \) |
| 3 | \( 1 + (1.52 - 0.819i)T \) |
| good | 5 | \( 1 + (-1.81 + 3.60i)T + (-2.98 - 4.01i)T^{2} \) |
| 7 | \( 1 + (-3.06 + 0.917i)T + (5.84 - 3.84i)T^{2} \) |
| 11 | \( 1 + (-0.0562 + 0.966i)T + (-10.9 - 1.27i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 1.35i)T + (-3.72 - 12.4i)T^{2} \) |
| 17 | \( 1 + (-1.25 - 1.49i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (4.84 - 5.76i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.64 + 5.50i)T + (-19.2 - 12.6i)T^{2} \) |
| 29 | \( 1 + (-3.97 - 1.71i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (-3.44 + 3.24i)T + (1.80 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-1.58 - 8.96i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (0.0921 - 0.788i)T + (-39.8 - 9.45i)T^{2} \) |
| 43 | \( 1 + (-0.0145 + 0.0220i)T + (-17.0 - 39.4i)T^{2} \) |
| 47 | \( 1 + (-4.87 + 5.16i)T + (-2.73 - 46.9i)T^{2} \) |
| 53 | \( 1 + (3.07 - 1.77i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.416 + 7.14i)T + (-58.6 + 6.84i)T^{2} \) |
| 61 | \( 1 + (-0.919 - 0.217i)T + (54.5 + 27.3i)T^{2} \) |
| 67 | \( 1 + (13.5 - 5.83i)T + (45.9 - 48.7i)T^{2} \) |
| 71 | \( 1 + (-3.54 - 1.29i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.192 + 0.0702i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.06 - 9.13i)T + (-76.8 + 18.2i)T^{2} \) |
| 83 | \( 1 + (3.99 - 0.466i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (-2.78 - 7.65i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (15.2 - 7.66i)T + (57.9 - 77.8i)T^{2} \) |
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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
−12.00730028770395663846388512879, −10.60290357438988010088847681015, −9.758843644547955102113100427092, −8.523610479245236351131673971611, −8.178693887574064340387709165717, −6.41700836484226193158254588790, −5.66022255832619104296619576067, −4.80784671028272412567300938825, −4.19491067286012872668922855233, −1.12153134774516041210921058527,
1.73678367446200826229437299646, 2.70184006227019188960833745020, 4.51757142311650401755929089395, 5.57167889640394172055509325937, 6.48190291829328171150919843965, 7.52289378181773616634762226256, 9.036167306389223302363122850935, 10.16403956817830330775808514427, 10.96344845696021187866341428780, 11.28636157142485769519918747904
