| L(s) = 1 | + (−0.503 + 1.21i)2-s + (0.802 − 1.53i)3-s + (0.192 + 0.192i)4-s + (−0.0781 + 0.116i)5-s + (1.46 + 1.74i)6-s + (−1.47 + 0.982i)7-s + (−2.75 + 1.14i)8-s + (−1.71 − 2.46i)9-s + (−0.102 − 0.153i)10-s + (0.710 − 3.57i)11-s + (0.450 − 0.141i)12-s + (−0.0825 + 0.0825i)13-s + (−0.453 − 2.28i)14-s + (0.116 + 0.213i)15-s − 3.38i·16-s + (−1.03 + 3.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.355 + 0.858i)2-s + (0.463 − 0.886i)3-s + (0.0962 + 0.0962i)4-s + (−0.0349 + 0.0523i)5-s + (0.596 + 0.712i)6-s + (−0.555 + 0.371i)7-s + (−0.975 + 0.404i)8-s + (−0.571 − 0.820i)9-s + (−0.0324 − 0.0486i)10-s + (0.214 − 1.07i)11-s + (0.129 − 0.0407i)12-s + (−0.0228 + 0.0228i)13-s + (−0.121 − 0.609i)14-s + (0.0301 + 0.0552i)15-s − 0.845i·16-s + (−0.251 + 0.967i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759288 + 0.227537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759288 + 0.227537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.802 + 1.53i)T \) |
| 17 | \( 1 + (1.03 - 3.99i)T \) |
| good | 2 | \( 1 + (0.503 - 1.21i)T + (-1.41 - 1.41i)T^{2} \) |
| 5 | \( 1 + (0.0781 - 0.116i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (1.47 - 0.982i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.710 + 3.57i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (0.0825 - 0.0825i)T - 13iT^{2} \) |
| 19 | \( 1 + (6.20 + 2.56i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-6.15 - 1.22i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.27 - 2.18i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-8.78 + 1.74i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-0.0726 - 0.365i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (2.44 + 3.65i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.94 + 0.806i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (2.11 + 2.11i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.45 - 5.91i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.83 + 3.65i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (2.20 + 3.30i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 5.81iT - 67T^{2} \) |
| 71 | \( 1 + (12.7 - 2.53i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (3.71 + 2.48i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.39 - 0.477i)T + (72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (5.11 + 2.12i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.89 - 2.89i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.54 - 3.81i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−15.54554210599555458246537145141, −14.78367001539534498903562831532, −13.39261473534130509644257331418, −12.43164478540328392458123592910, −11.13853421926958508720526368848, −8.984542540057268583668329713534, −8.349580044824016828883507954692, −6.88436008482150393144221567553, −6.08319236935005520725825323868, −3.00110358607803434111404687225,
2.71642285744915708695150566207, 4.48549138362951725071832122349, 6.65816285151518289113572907682, 8.654223790946049198978326825780, 9.849284661638447169925373212674, 10.42048108752056800331973008865, 11.72475318911069920034037355085, 12.98604539371198406553856975895, 14.51129020569494571155164483773, 15.38111822526255275730423064883
