| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.128 + 0.729i)5-s + (3.16 − 2.65i)7-s + (−0.500 − 0.866i)8-s + (0.370 − 0.641i)10-s + (0.789 + 4.47i)11-s + (−0.812 + 0.295i)13-s + (−3.88 + 1.41i)14-s + (0.173 + 0.984i)16-s + (2.17 − 3.77i)17-s + (−0.777 − 1.34i)19-s + (−0.567 + 0.476i)20-s + (0.789 − 4.47i)22-s + (−2.68 − 2.25i)23-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.0575 + 0.326i)5-s + (1.19 − 1.00i)7-s + (−0.176 − 0.306i)8-s + (0.117 − 0.202i)10-s + (0.238 + 1.34i)11-s + (−0.225 + 0.0820i)13-s + (−1.03 + 0.377i)14-s + (0.0434 + 0.246i)16-s + (0.528 − 0.915i)17-s + (−0.178 − 0.309i)19-s + (−0.126 + 0.106i)20-s + (0.168 − 0.954i)22-s + (−0.559 − 0.469i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18714 - 0.155051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18714 - 0.155051i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.128 - 0.729i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.16 + 2.65i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.789 - 4.47i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.812 - 0.295i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.17 + 3.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.777 + 1.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.68 + 2.25i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.07 - 1.11i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-8.19 - 6.87i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.42 + 0.881i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.478 + 2.71i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.42 + 6.23i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 + (-0.248 + 1.40i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.31 - 2.78i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (12.3 - 4.48i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.54 + 4.40i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.286 - 0.496i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.35 + 2.31i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (8.66 + 3.15i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.19 - 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.999 - 5.66i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.60793504404406633800831935884, −10.31665650540823073869779430663, −9.213465279767508717015735965932, −8.185963956864970133764456893427, −7.29837071445097533839021166712, −6.82567042172235403867530082897, −5.01125164390052608874145346786, −4.19922475740707594774547515094, −2.60269224730424647374628680295, −1.21958102987182044094430614607,
1.25433448165977799112592399548, 2.70610452099739922205854147483, 4.39198507085618966208289702233, 5.64209477056438635854801339569, 6.19356462751260877790025610401, 7.82959871578535045841809651070, 8.310140647811357363696182608668, 8.950805707581771557766841988800, 10.06917102924328527588301410705, 11.03088323503010619730149309305
