| L(s) = 1 | + (0.320 − 0.0421i)2-s + (−1.38 − 1.03i)3-s + (−1.83 + 0.490i)4-s + (−0.356 − 0.722i)5-s + (−0.487 − 0.273i)6-s + (−2.50 − 1.23i)7-s + (−1.16 + 0.481i)8-s + (0.850 + 2.87i)9-s + (−0.144 − 0.216i)10-s + (−4.03 + 1.37i)11-s + (3.04 + 1.21i)12-s + (−1.31 − 4.89i)13-s + (−0.853 − 0.289i)14-s + (−0.254 + 1.37i)15-s + (2.93 − 1.69i)16-s + (0.361 − 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.226 − 0.0297i)2-s + (−0.801 − 0.598i)3-s + (−0.915 + 0.245i)4-s + (−0.159 − 0.323i)5-s + (−0.199 − 0.111i)6-s + (−0.946 − 0.466i)7-s + (−0.410 + 0.170i)8-s + (0.283 + 0.958i)9-s + (−0.0456 − 0.0683i)10-s + (−1.21 + 0.413i)11-s + (0.880 + 0.351i)12-s + (−0.363 − 1.35i)13-s + (−0.228 − 0.0774i)14-s + (−0.0657 + 0.354i)15-s + (0.732 − 0.423i)16-s + (0.0877 − 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0352676 - 0.278427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0352676 - 0.278427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.38 + 1.03i)T \) |
| 17 | \( 1 + (-0.361 + 4.10i)T \) |
| good | 2 | \( 1 + (-0.320 + 0.0421i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (0.356 + 0.722i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (2.50 + 1.23i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (4.03 - 1.37i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (1.31 + 4.89i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-5.27 - 2.18i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (5.12 - 5.84i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.0416 + 0.634i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (0.702 - 2.06i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (0.897 + 4.51i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (4.17 - 0.273i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (6.50 + 4.99i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.793 - 2.96i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.67 - 8.86i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.28 + 9.74i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.138 - 0.280i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-2.17 - 1.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.92 + 0.383i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (4.62 + 3.08i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.95 + 5.76i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (1.40 + 10.6i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (9.50 - 9.50i)T - 89iT^{2} \) |
| 97 | \( 1 + (-11.4 - 0.753i)T + (96.1 + 12.6i)T^{2} \) |
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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
−12.61895227184428898240547328657, −11.92806372848123137895434710911, −10.33807283334022217156874176002, −9.723853586028232638588899055812, −8.023891275456941521861604600847, −7.35239294460619369774110840281, −5.65883823662650322370516608804, −4.94225195036725911496809884994, −3.21135944734395181152638816614, −0.27709563291459365619831205486,
3.33425923464663486850681425041, 4.65893958678836100461588888345, 5.70765170339332252449202784764, 6.71283720926649491564434220132, 8.512487433947761220378274820275, 9.626441552159841948951312217063, 10.23627155527036473761563317154, 11.46303524049715610183647786729, 12.48409733491570395145538303066, 13.31254089569548406134772614344
