| 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
−11.23377466607366163325435878595, −10.86428146105198055730166759677, −9.444205173474731402072302292979, −8.940156832169907119505316021979, −7.72552035723391060516590021049, −6.97057444343241902180486363937, −5.22942356399035232667395988169, −4.07149239702369732164103918054, −3.92694871249345919497894881164, −2.59455803169582480900135172835,
0.68895164530762975533888397334, 2.78083944524896837214263085667, 3.78943474862056142404696059821, 5.34824004660800611099920514492, 5.88992763965052175654658394192, 7.12614054219148196967790996783, 7.977345773340016549804983341763, 8.913442850821346343693921771750, 9.678656352468223986184856056887, 11.02617693264931451393247889275
