| L(s) = 1 | + (−1.00 − 0.644i)2-s + (0.959 + 0.281i)3-s + (−0.241 − 0.527i)4-s + (−1.03 + 1.19i)5-s + (−0.780 − 0.900i)6-s + (3.81 + 2.45i)7-s + (−0.437 + 3.04i)8-s + (0.841 + 0.540i)9-s + (1.80 − 0.530i)10-s + (3.64 − 4.20i)11-s + (−0.0825 − 0.574i)12-s + (−0.313 − 2.17i)13-s + (−2.24 − 4.91i)14-s + (−1.32 + 0.853i)15-s + (1.63 − 1.89i)16-s + (−2.08 + 4.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.708 − 0.455i)2-s + (0.553 + 0.162i)3-s + (−0.120 − 0.263i)4-s + (−0.462 + 0.533i)5-s + (−0.318 − 0.367i)6-s + (1.44 + 0.926i)7-s + (−0.154 + 1.07i)8-s + (0.280 + 0.180i)9-s + (0.571 − 0.167i)10-s + (1.09 − 1.26i)11-s + (−0.0238 − 0.165i)12-s + (−0.0869 − 0.604i)13-s + (−0.599 − 1.31i)14-s + (−0.343 + 0.220i)15-s + (0.409 − 0.472i)16-s + (−0.506 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02090 - 0.128938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02090 - 0.128938i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (3.95 - 7.16i)T \) |
| good | 2 | \( 1 + (1.00 + 0.644i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.03 - 1.19i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-3.81 - 2.45i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-3.64 + 4.20i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.313 + 2.17i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.08 - 4.57i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.18 + 3.32i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (4.65 + 1.36i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + (-0.206 + 1.43i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + (1.76 - 3.85i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (3.28 - 7.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (1.64 + 0.481i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (3.39 + 7.43i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.00846 - 0.0588i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (9.65 + 11.1i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (3.12 + 6.84i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (5.05 + 5.83i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.755 + 5.25i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.76 + 5.49i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.70 + 1.38i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−11.86355528005871885566754980053, −11.34102537067594546252678156303, −10.56655193853290565353946926160, −9.280865812651797823075108936859, −8.509282964994291063896916428175, −7.926014464364287857012278555877, −6.11326526098405217858655035590, −4.87499544764696477592803552864, −3.20258229440169271739155662963, −1.64445174941750380243373216670,
1.46101924265207771609763919649, 3.95430469774635508337382286890, 4.66291194908959279843775446633, 7.00545997929661043305933447331, 7.48896897327258409709720713941, 8.398864212416526443170239125774, 9.249757978518895542431494544870, 10.19122919202441890098341666092, 11.89370255789972977676502215484, 12.12322396938167009831910262892
