| L(s) = 1 | + (−0.395 + 1.73i)2-s + (1.72 + 0.196i)3-s + (−1.03 − 0.500i)4-s + (1.06 + 2.71i)5-s + (−1.02 + 2.90i)6-s + (−2.62 + 0.301i)7-s + (−0.937 + 1.17i)8-s + (2.92 + 0.677i)9-s + (−5.11 + 0.771i)10-s + (2.00 − 0.619i)11-s + (−1.68 − 1.06i)12-s + (−2.86 + 0.884i)13-s + (0.515 − 4.66i)14-s + (1.29 + 4.87i)15-s + (−3.10 − 3.89i)16-s + (0.544 − 7.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.279 + 1.22i)2-s + (0.993 + 0.113i)3-s + (−0.519 − 0.250i)4-s + (0.476 + 1.21i)5-s + (−0.416 + 1.18i)6-s + (−0.993 + 0.114i)7-s + (−0.331 + 0.415i)8-s + (0.974 + 0.225i)9-s + (−1.61 + 0.243i)10-s + (0.605 − 0.186i)11-s + (−0.487 − 0.307i)12-s + (−0.795 + 0.245i)13-s + (0.137 − 1.24i)14-s + (0.335 + 1.25i)15-s + (−0.775 − 0.972i)16-s + (0.132 − 1.76i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.364124 + 1.62533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364124 + 1.62533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.196i)T \) |
| 7 | \( 1 + (2.62 - 0.301i)T \) |
| good | 2 | \( 1 + (0.395 - 1.73i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (-1.06 - 2.71i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-2.00 + 0.619i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (2.86 - 0.884i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (-0.544 + 7.27i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.87 + 1.96i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (0.501 - 6.69i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + (1.73 + 1.18i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-2.66 - 0.401i)T + (39.1 + 12.0i)T^{2} \) |
| 43 | \( 1 + (4.80 - 0.723i)T + (41.0 - 12.6i)T^{2} \) |
| 47 | \( 1 + (2.41 - 10.5i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 7.70i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (8.56 + 10.7i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-1.85 + 0.895i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + (-7.34 - 3.53i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.80 + 0.556i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + 6.64T + 79T^{2} \) |
| 83 | \( 1 + (3.28 + 1.01i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.804i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (1.86 - 3.23i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.42588875725035371487515580501, −10.12563210711056228797697471747, −9.502437281406893646950376470139, −8.817611318162085165700571224977, −7.52777301193717457922487399354, −6.95682641258405212214211358627, −6.38429363115959623408088141505, −5.02226135656800125134205670502, −3.24652641544518801859653784504, −2.58442845363304850503619662659,
1.10082668912209059042360841855, 2.25629484319207869197142689974, 3.43882535174317434086153159783, 4.40424984709966073971297854899, 6.01532618352159466120788571688, 7.15615762412864078471119086939, 8.536000054690797578472515440109, 9.128515053457526316330298281649, 9.859413152836693090235406501238, 10.34221792545185621558621572514
