| 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
−10.70111505598932622237773454447, −9.441230571549330811705717523776, −8.757503974726403944138482369775, −8.010160754765619624216276946951, −6.89911928015032437836930091210, −5.54361729655722130458877194582, −4.82526419880881362444051952003, −4.29422544905455761499982613800, −2.89814604064344455828142901284, −1.68475625065057087344874778634,
1.00415111810951139295098173432, 3.27037687333684196147575573079, 3.63018272960433066693562651135, 5.07172410766409037885572274729, 5.73002739886537774799115562035, 6.59301207249055632953795093993, 7.64030319586250636816337596084, 8.415551952473219446843119921299, 9.544076401918778185432439829415, 10.58211041567997647939209598285
