L(s) = 1 | + (1.40 + 0.112i)2-s + (1.97 + 0.316i)4-s + (−2.18 − 0.466i)5-s + 1.00·7-s + (2.74 + 0.667i)8-s + (−3.03 − 0.903i)10-s + (1.89 − 1.89i)11-s + (2.65 − 2.65i)13-s + (1.40 + 0.112i)14-s + (3.80 + 1.24i)16-s − 1.73i·17-s + (5.33 + 5.33i)19-s + (−4.17 − 1.61i)20-s + (2.88 − 2.45i)22-s − 0.160·23-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0792i)2-s + (0.987 + 0.158i)4-s + (−0.977 − 0.208i)5-s + 0.378·7-s + (0.971 + 0.235i)8-s + (−0.958 − 0.285i)10-s + (0.571 − 0.571i)11-s + (0.737 − 0.737i)13-s + (0.376 + 0.0299i)14-s + (0.950 + 0.312i)16-s − 0.421i·17-s + (1.22 + 1.22i)19-s + (−0.932 − 0.360i)20-s + (0.614 − 0.524i)22-s − 0.0334·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.73868 - 0.185265i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.73868 - 0.185265i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.112i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.18 + 0.466i)T \) |
good | 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 - 5.33i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 + (2.70 + 2.70i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (7.23 + 7.23i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (3.44 + 3.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.101 - 0.101i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.01 + 6.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9.04 + 9.04i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.60iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (2.04 - 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 3.84iT - 97T^{2} \) |
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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
−10.76247877107416447055384405350, −9.615630207685895371865346157122, −8.207157332023130367951804763738, −7.889706585592441998640732285049, −6.77306879153231760835132979819, −5.76297090108261491871687686942, −4.92645532729498318119044757232, −3.77318521755532359081465124204, −3.21464803256148302301034849215, −1.33771878705320292523327577761,
1.54118266093905306944235028752, 3.06282762064762928663835972204, 4.03086682404993034914176569177, 4.72631052523084270222095865635, 5.89249605716956193657595795025, 7.01794398576749101314593510148, 7.42852950540988902722320816856, 8.633330091219513869716287819662, 9.634773042108391263063056792260, 11.00720613598875711389480361334