| L(s) = 1 | + (1.04 + 0.713i)2-s + (−0.733 − 0.680i)3-s + (−0.143 − 0.366i)4-s + (1.23 − 0.382i)5-s + (−0.282 − 1.23i)6-s + (2.59 − 0.536i)7-s + (0.675 − 2.95i)8-s + (0.0747 + 0.997i)9-s + (1.56 + 0.484i)10-s + (−0.384 + 5.12i)11-s + (−0.143 + 0.366i)12-s + (−1.76 + 0.852i)13-s + (3.09 + 1.28i)14-s + (−1.16 − 0.562i)15-s + (2.24 − 2.07i)16-s + (−4.91 + 0.740i)17-s + ⋯ |
| L(s) = 1 | + (0.740 + 0.504i)2-s + (−0.423 − 0.392i)3-s + (−0.0719 − 0.183i)4-s + (0.553 − 0.170i)5-s + (−0.115 − 0.504i)6-s + (0.979 − 0.202i)7-s + (0.238 − 1.04i)8-s + (0.0249 + 0.332i)9-s + (0.496 + 0.153i)10-s + (−0.115 + 1.54i)11-s + (−0.0415 + 0.105i)12-s + (−0.490 + 0.236i)13-s + (0.827 + 0.344i)14-s + (−0.301 − 0.145i)15-s + (0.560 − 0.519i)16-s + (−1.19 + 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49797 - 0.0784923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49797 - 0.0784923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.733 + 0.680i)T \) |
| 7 | \( 1 + (-2.59 + 0.536i)T \) |
| good | 2 | \( 1 + (-1.04 - 0.713i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (-1.23 + 0.382i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.384 - 5.12i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.852i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (4.91 - 0.740i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (3.33 + 5.76i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.69 - 0.708i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-4.28 - 5.37i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (0.371 - 0.643i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.11 - 7.93i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (1.45 - 6.39i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (2.60 + 11.4i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-2.15 - 1.46i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (0.584 + 1.48i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (3.40 + 1.05i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.0708 + 0.180i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.84 + 6.65i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.49 + 4.37i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.28 + 1.55i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (0.398 + 0.690i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.92 - 1.40i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.432 + 5.77i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−13.20968368944758685878789514146, −12.35701832847780449584489880400, −11.08996223935134528896473574630, −10.08393385767504764498572970999, −8.894454312872722363666815811802, −7.21800687701175439844457447653, −6.60974698454645855278110219655, −4.97746410218018208142903710320, −4.69414456225151305683645024676, −1.85481549254523841838719943945,
2.42271754162015809011403585944, 3.99892209955797728983028804207, 5.15836052086234107911224511398, 6.12273902476848312635633418164, 8.004768284887383928139618417529, 8.886807416327372627210038676775, 10.49292050081943685030847746834, 11.16989094058682742963476007063, 11.99047314451948668968129112297, 13.08135037875487670863771844493
