| L(s) = 1 | + (1.26 + 0.460i)2-s + (−0.141 − 0.118i)4-s + (−0.286 + 1.62i)5-s + (1.84 − 1.55i)7-s + (−1.47 − 2.54i)8-s + (−1.11 + 1.92i)10-s + (−1.03 − 5.85i)11-s + (3.03 − 1.10i)13-s + (3.05 − 1.11i)14-s + (−0.624 − 3.54i)16-s + (1.5 − 2.59i)17-s + (3.31 + 5.74i)19-s + (0.233 − 0.196i)20-s + (1.39 − 7.88i)22-s + (2.25 + 1.89i)23-s + ⋯ |
| L(s) = 1 | + (0.895 + 0.325i)2-s + (−0.0707 − 0.0593i)4-s + (−0.128 + 0.727i)5-s + (0.698 − 0.585i)7-s + (−0.520 − 0.901i)8-s + (−0.352 + 0.609i)10-s + (−0.311 − 1.76i)11-s + (0.840 − 0.306i)13-s + (0.815 − 0.296i)14-s + (−0.156 − 0.885i)16-s + (0.363 − 0.630i)17-s + (0.761 + 1.31i)19-s + (0.0523 − 0.0438i)20-s + (0.296 − 1.68i)22-s + (0.470 + 0.394i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.20050 - 0.521530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20050 - 0.521530i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.26 - 0.460i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.286 - 1.62i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.84 + 1.55i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (1.03 + 5.85i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-3.03 + 1.10i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 - 5.74i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.25 - 1.89i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.21 + 0.441i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.450 + 0.378i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.0209 - 0.0362i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.60 - 1.67i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.900 + 5.10i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.86 + 2.40i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-1.27 + 7.23i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.46 + 7.10i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.74 - 0.635i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.75 - 4.77i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 + 4.81i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.55 - 1.29i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.74 - 1.36i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.07 - 7.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0452 - 0.256i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.58211041567997647939209598285, −9.544076401918778185432439829415, −8.415551952473219446843119921299, −7.64030319586250636816337596084, −6.59301207249055632953795093993, −5.73002739886537774799115562035, −5.07172410766409037885572274729, −3.63018272960433066693562651135, −3.27037687333684196147575573079, −1.00415111810951139295098173432,
1.68475625065057087344874778634, 2.89814604064344455828142901284, 4.29422544905455761499982613800, 4.82526419880881362444051952003, 5.54361729655722130458877194582, 6.89911928015032437836930091210, 8.010160754765619624216276946951, 8.757503974726403944138482369775, 9.441230571549330811705717523776, 10.70111505598932622237773454447
