| L(s) = 1 | + (0.893 + 0.430i)2-s + (0.804 − 1.53i)3-s + (−0.633 − 0.794i)4-s + (−1.72 − 0.533i)5-s + (1.37 − 1.02i)6-s + (−0.957 + 2.46i)7-s + (−0.665 − 2.91i)8-s + (−1.70 − 2.46i)9-s + (−1.31 − 1.22i)10-s + (−0.0695 − 0.928i)11-s + (−1.72 + 0.332i)12-s + (−0.415 − 5.54i)13-s + (−1.91 + 1.79i)14-s + (−2.20 + 2.22i)15-s + (0.208 − 0.912i)16-s + (−1.33 − 3.41i)17-s + ⋯ |
| L(s) = 1 | + (0.631 + 0.304i)2-s + (0.464 − 0.885i)3-s + (−0.316 − 0.397i)4-s + (−0.773 − 0.238i)5-s + (0.562 − 0.418i)6-s + (−0.361 + 0.932i)7-s + (−0.235 − 1.03i)8-s + (−0.568 − 0.822i)9-s + (−0.416 − 0.386i)10-s + (−0.0209 − 0.279i)11-s + (−0.498 + 0.0961i)12-s + (−0.115 − 1.53i)13-s + (−0.512 + 0.478i)14-s + (−0.570 + 0.574i)15-s + (0.0520 − 0.228i)16-s + (−0.324 − 0.827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.724396 - 1.18114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.724396 - 1.18114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.804 + 1.53i)T \) |
| 7 | \( 1 + (0.957 - 2.46i)T \) |
| good | 2 | \( 1 + (-0.893 - 0.430i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (1.72 + 0.533i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0695 + 0.928i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (0.415 + 5.54i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (1.33 + 3.41i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.79 - 4.84i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.31 + 0.801i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-0.146 - 0.373i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 + (0.0638 + 0.00962i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-7.06 + 6.55i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-9.16 - 8.50i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-3.33 - 1.60i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-12.1 + 1.83i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-1.30 + 5.72i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.609 - 0.763i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 + (-2.26 - 2.84i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.211 - 2.82i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + (-0.822 + 10.9i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (9.58 + 6.53i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-1.03 + 1.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.02617693264931451393247889275, −9.678656352468223986184856056887, −8.913442850821346343693921771750, −7.977345773340016549804983341763, −7.12614054219148196967790996783, −5.88992763965052175654658394192, −5.34824004660800611099920514492, −3.78943474862056142404696059821, −2.78083944524896837214263085667, −0.68895164530762975533888397334,
2.59455803169582480900135172835, 3.92694871249345919497894881164, 4.07149239702369732164103918054, 5.22942356399035232667395988169, 6.97057444343241902180486363937, 7.72552035723391060516590021049, 8.940156832169907119505316021979, 9.444205173474731402072302292979, 10.86428146105198055730166759677, 11.23377466607366163325435878595
