| L(s) = 1 | + (0.399 − 0.916i)2-s + (1.04 + 1.37i)3-s + (−0.680 − 0.733i)4-s + (0.292 + 0.0440i)5-s + (1.68 − 0.410i)6-s + (1.79 + 1.66i)7-s + (−0.943 + 0.330i)8-s + (−0.800 + 2.89i)9-s + (0.157 − 0.250i)10-s + (−2.06 + 1.77i)11-s + (0.297 − 1.70i)12-s + (3.14 − 4.61i)13-s + (2.24 − 0.978i)14-s + (0.245 + 0.449i)15-s + (−0.0747 + 0.997i)16-s + (3.95 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (0.282 − 0.648i)2-s + (0.605 + 0.795i)3-s + (−0.340 − 0.366i)4-s + (0.130 + 0.0197i)5-s + (0.687 − 0.167i)6-s + (0.677 + 0.629i)7-s + (−0.333 + 0.116i)8-s + (−0.266 + 0.963i)9-s + (0.0497 − 0.0791i)10-s + (−0.623 + 0.536i)11-s + (0.0857 − 0.492i)12-s + (0.872 − 1.27i)13-s + (0.599 − 0.261i)14-s + (0.0635 + 0.116i)15-s + (−0.0186 + 0.249i)16-s + (0.959 + 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07426 + 0.265778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07426 + 0.265778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.399 + 0.916i)T \) |
| 3 | \( 1 + (-1.04 - 1.37i)T \) |
| 29 | \( 1 + (5.30 - 0.936i)T \) |
| good | 5 | \( 1 + (-0.292 - 0.0440i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-1.79 - 1.66i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (2.06 - 1.77i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.14 + 4.61i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-3.95 - 3.95i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.86 - 1.17i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 1.73i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 0.958i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (2.89 + 8.26i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-12.2 - 3.27i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.75 - 2.37i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (4.33 + 5.03i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (9.81 + 7.82i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-0.938 + 0.541i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-13.9 + 0.520i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (5.41 - 0.405i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (8.05 - 3.87i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.392 + 3.48i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-2.05 + 10.8i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (1.97 - 6.39i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (5.48 - 0.618i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (5.12 + 9.69i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.81720784031158603380867082369, −10.11566759880212840468782309853, −9.335570579737710777338050730676, −8.273060758946358125404782347538, −7.75462041501026713139749607389, −5.68514828668946541417696079583, −5.29101370291475206159282195528, −3.95718595584035449262990896532, −3.04831384364604377355063244306, −1.81743768460202075362266726233,
1.26529509497066286143881848246, 2.93197420675279802967814448256, 4.09907171998372355588743326650, 5.34338266798419350902029268277, 6.38902657694275213321938063305, 7.34566220936066359657468790654, 7.88433831743525116382056530966, 8.856203115762917426193473104254, 9.610983585781591201622228571226, 11.09206119071118505391006602612
