| L(s) = 1 | + (1.25 − 0.655i)2-s + (0.654 + 0.755i)3-s + (1.13 − 1.64i)4-s + (−0.329 − 2.29i)5-s + (1.31 + 0.517i)6-s + (−0.102 + 0.223i)7-s + (0.349 − 2.80i)8-s + (−0.142 + 0.989i)9-s + (−1.91 − 2.65i)10-s + (−1.07 + 3.67i)11-s + (1.98 − 0.215i)12-s + (4.96 − 2.26i)13-s + (0.0187 + 0.346i)14-s + (1.51 − 1.75i)15-s + (−1.40 − 3.74i)16-s + (3.16 − 4.92i)17-s + ⋯ |
| L(s) = 1 | + (0.885 − 0.463i)2-s + (0.378 + 0.436i)3-s + (0.569 − 0.821i)4-s + (−0.147 − 1.02i)5-s + (0.537 + 0.211i)6-s + (−0.0385 + 0.0844i)7-s + (0.123 − 0.992i)8-s + (−0.0474 + 0.329i)9-s + (−0.605 − 0.839i)10-s + (−0.325 + 1.10i)11-s + (0.574 − 0.0620i)12-s + (1.37 − 0.629i)13-s + (0.00499 + 0.0927i)14-s + (0.391 − 0.451i)15-s + (−0.350 − 0.936i)16-s + (0.767 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28467 - 1.42120i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28467 - 1.42120i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.25 + 0.655i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.134 - 4.79i)T \) |
| good | 5 | \( 1 + (0.329 + 2.29i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.102 - 0.223i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.07 - 3.67i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.96 + 2.26i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.16 + 4.92i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.56 + 3.98i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (5.43 - 8.46i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.84 + 2.46i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.0873 + 0.607i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 7.51i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.07 - 3.53i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.48iT - 47T^{2} \) |
| 53 | \( 1 + (-3.61 + 7.92i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.19 + 4.80i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.00 - 2.31i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.858 - 2.92i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.30 - 14.6i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.82 + 5.66i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.27 - 4.97i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.0564 - 0.00811i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (12.4 - 10.8i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (15.6 - 2.25i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.83915898005710086088098463729, −9.691287432146872187310019425526, −9.168917005683294025854938914208, −7.984641357986162700795768389658, −6.94202106567648722328241736047, −5.50110338760624607928174431945, −4.95293672789931729694424724519, −3.94175318017194527378348795700, −2.86961873675824487190157819825, −1.33000483477582806426896773814,
2.07454043713770189939550390326, 3.45234635420566877391203527514, 3.88962291927369828165817825810, 5.80647803404673729480253334307, 6.23438888801962040093463948007, 7.18461595206943340419271859326, 8.176980542551476639607958814110, 8.690866612199763998900823480547, 10.48416022191269566535182394163, 10.95687573930938769503437963759
