| L(s) = 1 | + (−0.318 − 0.267i)2-s + (0.159 + 1.72i)3-s + (−0.317 − 1.79i)4-s + (−2.08 − 0.757i)5-s + (0.409 − 0.591i)6-s + (−0.229 + 1.29i)7-s + (−0.795 + 1.37i)8-s + (−2.94 + 0.551i)9-s + (0.460 + 0.797i)10-s + (4.90 − 1.78i)11-s + (3.05 − 0.834i)12-s + (−0.0138 + 0.0116i)13-s + (0.419 − 0.352i)14-s + (0.974 − 3.71i)15-s + (−2.81 + 1.02i)16-s + (1.56 + 2.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 − 0.188i)2-s + (0.0922 + 0.995i)3-s + (−0.158 − 0.899i)4-s + (−0.930 − 0.338i)5-s + (0.167 − 0.241i)6-s + (−0.0866 + 0.491i)7-s + (−0.281 + 0.486i)8-s + (−0.982 + 0.183i)9-s + (0.145 + 0.252i)10-s + (1.47 − 0.537i)11-s + (0.881 − 0.241i)12-s + (−0.00383 + 0.00321i)13-s + (0.112 − 0.0941i)14-s + (0.251 − 0.958i)15-s + (−0.703 + 0.256i)16-s + (0.379 + 0.658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.567882 + 0.0126348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.567882 + 0.0126348i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.159 - 1.72i)T \) |
| good | 2 | \( 1 + (0.318 + 0.267i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (2.08 + 0.757i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.229 - 1.29i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.90 + 1.78i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0138 - 0.0116i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 2.71i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.208 - 0.361i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.179 + 1.01i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.98 + 5.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.647 - 3.67i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.21 + 3.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.80 + 2.84i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.23 - 6.99i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 + (-3.47 - 1.26i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.44 - 7.08i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.273 + 0.473i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.374 - 0.314i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.53 - 2.96i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.68 + 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.34 - 3.40i)T + (74.3 - 62.3i)T^{2} \) |
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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
−17.26757666820539469153793665031, −16.05964725446672923597591682014, −15.06511953194407883224456892518, −14.18179142293265801263626882678, −11.97756399136526140300114527250, −10.96416714558952203217923774030, −9.505094398920335895429043131524, −8.534705305699404093866563920772, −5.86274127825850910906463735296, −4.07348243576516157456316411183,
3.68771838975598762116168332143, 6.89586373664414977827306987784, 7.64598089807649843595038236107, 9.110140218839915662927826793138, 11.51306656287596393333514178044, 12.26240124723143498264230542380, 13.60434301001082601018435284033, 14.91361505773453857725970005165, 16.54837303970361264501705096268, 17.40248005733723778142719794333
