| L(s) = 1 | + (0.651 − 1.25i)2-s + (−1.44 − 0.598i)3-s + (−1.15 − 1.63i)4-s + (2.07 − 1.05i)5-s + (−1.69 + 1.42i)6-s + (−0.213 − 0.348i)7-s + (−2.80 + 0.381i)8-s + (−0.392 − 0.392i)9-s + (0.0238 − 3.29i)10-s + (−1.39 − 0.109i)11-s + (0.685 + 3.05i)12-s + (−0.0467 − 0.194i)13-s + (−0.576 + 0.0411i)14-s + (−3.63 + 0.286i)15-s + (−1.34 + 3.76i)16-s + (1.14 − 0.982i)17-s + ⋯ |
| L(s) = 1 | + (0.460 − 0.887i)2-s + (−0.834 − 0.345i)3-s + (−0.576 − 0.817i)4-s + (0.929 − 0.473i)5-s + (−0.690 + 0.581i)6-s + (−0.0807 − 0.131i)7-s + (−0.990 + 0.134i)8-s + (−0.130 − 0.130i)9-s + (0.00755 − 1.04i)10-s + (−0.419 − 0.0330i)11-s + (0.198 + 0.880i)12-s + (−0.0129 − 0.0540i)13-s + (−0.154 + 0.0110i)14-s + (−0.939 + 0.0739i)15-s + (−0.336 + 0.941i)16-s + (0.278 − 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.439219 - 1.01823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439219 - 1.01823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.651 + 1.25i)T \) |
| 41 | \( 1 + (3.12 - 5.58i)T \) |
| good | 3 | \( 1 + (1.44 + 0.598i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.07 + 1.05i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.213 + 0.348i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (1.39 + 0.109i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.0467 + 0.194i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 0.982i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-5.19 - 1.24i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-7.30 + 5.31i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.29 - 4.52i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (0.798 + 2.45i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.0154 - 0.0474i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (6.26 + 0.992i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.42 + 3.96i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (3.80 - 4.45i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-1.05 - 1.44i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.16 - 1.45i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.256 - 3.26i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (1.06 - 13.5i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-5.44 + 5.44i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.94 - 4.70i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 2.69iT - 83T^{2} \) |
| 89 | \( 1 + (2.20 - 1.35i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (1.30 + 16.5i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.46003973137357606525732810315, −11.62156430352567828052378560561, −10.63099706408011003863833212891, −9.720875318828700744559969934864, −8.713405118432237528077833252123, −6.78855034425809287749041317016, −5.62467869328917157430630551956, −4.96988120176438844404823860748, −3.00714954055636929041877981701, −1.13019887466151999888438900509,
3.02438840509189612059346805762, 4.92127738210468952172040511315, 5.63282310089500338623970633595, 6.57699659497226639294208325667, 7.76847035587862222268586032343, 9.186040070819005648814783297466, 10.16952178558923897806872340142, 11.29219909631085432744978840810, 12.30423009549328217878539793332, 13.54816488838485623093492602500
