| L(s) = 1 | + (1.06 − 0.140i)2-s + (−0.442 − 0.896i)3-s + (−0.812 + 0.217i)4-s + (−2.05 + 1.80i)5-s + (−0.598 − 0.895i)6-s + (−0.418 − 2.61i)7-s + (−2.82 + 1.17i)8-s + (−0.608 + 0.793i)9-s + (−1.93 + 2.21i)10-s + (−1.62 + 0.106i)11-s + (0.554 + 0.632i)12-s + (−3.58 − 3.58i)13-s + (−0.813 − 2.72i)14-s + (2.52 + 1.04i)15-s + (−1.39 + 0.805i)16-s + (−3.46 + 2.24i)17-s + ⋯ |
| L(s) = 1 | + (0.754 − 0.0993i)2-s + (−0.255 − 0.517i)3-s + (−0.406 + 0.108i)4-s + (−0.918 + 0.805i)5-s + (−0.244 − 0.365i)6-s + (−0.158 − 0.987i)7-s + (−0.999 + 0.413i)8-s + (−0.202 + 0.264i)9-s + (−0.613 + 0.699i)10-s + (−0.489 + 0.0321i)11-s + (0.160 + 0.182i)12-s + (−0.995 − 0.995i)13-s + (−0.217 − 0.729i)14-s + (0.651 + 0.269i)15-s + (−0.348 + 0.201i)16-s + (−0.839 + 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00634644 + 0.126352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00634644 + 0.126352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.442 + 0.896i)T \) |
| 7 | \( 1 + (0.418 + 2.61i)T \) |
| 17 | \( 1 + (3.46 - 2.24i)T \) |
| good | 2 | \( 1 + (-1.06 + 0.140i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (2.05 - 1.80i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (1.62 - 0.106i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (3.58 + 3.58i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.0797 - 0.605i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-2.88 - 1.42i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.0592 - 0.298i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (2.75 - 1.35i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.441 + 6.73i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-2.11 + 10.6i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.59 - 8.68i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (0.122 - 0.458i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.67 - 8.70i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (6.53 + 0.860i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (2.28 + 6.71i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (9.18 + 5.30i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.90 + 3.94i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (13.8 + 4.70i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 8.01i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (4.06 - 9.81i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (6.14 + 1.64i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-15.6 + 3.11i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.99877833145001230281006669471, −10.49404217834024343397063967160, −9.069999700604281381084700190977, −7.69639754417552586718267345928, −7.35702057883084513956035730543, −6.04629854036808717194725111115, −4.85323606680837761842441035386, −3.81081380992430078729998539456, −2.81022415855200672066844540740, −0.06762353179832924984477400321,
2.83376269890012931572907906563, 4.35039350819025742675750895015, 4.77743060886346755793311828229, 5.74329428840991592265846705032, 7.01714956379197254925770515912, 8.512362100089474425055163290710, 9.071190928201626309640637811676, 9.920083176937730849665421435064, 11.45940230653927384547006978639, 11.95839581683901889287385964531
