| L(s) = 1 | + (1.05 + 0.294i)2-s + (1.61 − 0.623i)3-s + (−0.681 − 0.411i)4-s + (−0.410 + 0.0159i)5-s + (1.89 − 0.183i)6-s + (3.12 + 0.488i)7-s + (−2.10 − 2.23i)8-s + (2.22 − 2.01i)9-s + (−0.438 − 0.103i)10-s + (−0.0946 − 0.693i)11-s + (−1.35 − 0.239i)12-s + (−3.19 + 4.65i)13-s + (3.15 + 1.43i)14-s + (−0.653 + 0.281i)15-s + (−0.826 − 1.56i)16-s + (3.16 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (0.747 + 0.208i)2-s + (0.932 − 0.359i)3-s + (−0.340 − 0.205i)4-s + (−0.183 + 0.00712i)5-s + (0.772 − 0.0748i)6-s + (1.18 + 0.184i)7-s + (−0.744 − 0.788i)8-s + (0.740 − 0.671i)9-s + (−0.138 − 0.0328i)10-s + (−0.0285 − 0.208i)11-s + (−0.391 − 0.0692i)12-s + (−0.886 + 1.29i)13-s + (0.844 + 0.383i)14-s + (−0.168 + 0.0727i)15-s + (−0.206 − 0.392i)16-s + (0.766 − 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.10279 - 0.302767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.10279 - 0.302767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.61 + 0.623i)T \) |
| good | 2 | \( 1 + (-1.05 - 0.294i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (0.410 - 0.0159i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-3.12 - 0.488i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.0946 + 0.693i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (3.19 - 4.65i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-3.16 + 1.58i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.0397 - 0.681i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (2.18 - 5.65i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (5.86 + 4.19i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.26 - 3.74i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.89 - 2.54i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 5.30i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-8.75 - 7.94i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (8.82 + 10.1i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.850 - 4.82i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.80 + 2.37i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (0.455 - 0.274i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-0.213 + 0.152i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (0.202 + 0.678i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.98 - 1.18i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 10.4i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.62 - 6.29i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-2.51 + 8.39i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-9.95 - 0.386i)T + (96.7 + 7.51i)T^{2} \) |
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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
−12.14873103060307619411650903316, −11.52888534897462849097575413246, −9.797656759438240407112022429901, −9.194289234553984339893917808632, −8.000228450739912193543110829841, −7.20177205987049303743464134548, −5.75491209370417918289096750478, −4.61481801675108799334957471386, −3.61905607073902637853812504473, −1.85874419630659742937270246232,
2.34416976683006072431965727703, 3.65352991982076315152033103707, 4.61676853923605101286141704923, 5.51149148366794244862211024222, 7.74020101374070470308721973114, 7.985731023111704783352713759052, 9.194996966228631674047466959714, 10.25901559717099623155925559695, 11.27990023306966573242983900638, 12.49104115655605406232492585024
