| L(s) = 1 | + (1.28 + 0.598i)2-s + (−1.22 − 0.243i)3-s + (1.28 + 1.53i)4-s + (0.884 − 1.32i)5-s + (−1.41 − 1.04i)6-s + (−2.40 + 0.997i)7-s + (0.723 + 2.73i)8-s + (−1.33 − 0.554i)9-s + (1.92 − 1.16i)10-s + (−0.432 − 2.17i)11-s + (−1.19 − 2.18i)12-s + (−2.18 − 3.26i)13-s + (−3.68 − 0.164i)14-s + (−1.40 + 1.40i)15-s + (−0.710 + 3.93i)16-s + (4.38 + 4.38i)17-s + ⋯ |
| L(s) = 1 | + (0.905 + 0.423i)2-s + (−0.705 − 0.140i)3-s + (0.641 + 0.767i)4-s + (0.395 − 0.592i)5-s + (−0.579 − 0.425i)6-s + (−0.909 + 0.376i)7-s + (0.255 + 0.966i)8-s + (−0.445 − 0.184i)9-s + (0.609 − 0.368i)10-s + (−0.130 − 0.655i)11-s + (−0.344 − 0.631i)12-s + (−0.605 − 0.905i)13-s + (−0.983 − 0.0439i)14-s + (−0.362 + 0.362i)15-s + (−0.177 + 0.984i)16-s + (1.06 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10649 + 0.235564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10649 + 0.235564i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.598i)T \) |
| good | 3 | \( 1 + (1.22 + 0.243i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-0.884 + 1.32i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (2.40 - 0.997i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.432 + 2.17i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (2.18 + 3.26i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.38 - 4.38i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.64 + 1.76i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (2.12 - 5.12i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (0.836 - 4.20i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + 8.37iT - 31T^{2} \) |
| 37 | \( 1 + (5.42 + 3.62i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-3.00 + 7.24i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-8.60 + 1.71i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.0771 - 0.0771i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.846 - 4.25i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (0.657 - 0.984i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (9.90 + 1.97i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (9.06 + 1.80i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-9.94 + 4.12i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (10.8 + 4.48i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (0.842 - 0.842i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.766 + 0.512i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-2.03 - 4.90i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 5.90iT - 97T^{2} \) |
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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
−15.05679987713507849382122366562, −13.77811464709900170777725059727, −12.70868396613879635768211808070, −12.14525849489278590350087494475, −10.78680710266815247613740035859, −9.148852221429236157271252965191, −7.59388839219757000291529558130, −5.86744853729931808019686081943, −5.52995393576870715655751278696, −3.28101430132930729548649170246,
2.85509742321192189215332966903, 4.74978254809057322288729408702, 6.11297537110601770140549621857, 7.11289923838152877031413075509, 9.764861992049667037015815641716, 10.40023905413113979179782578892, 11.72718028463330889053498469629, 12.46297435820463202598837951451, 13.95695798270517102492763124681, 14.42286941831930070684838924376
