| L(s) = 1 | + (0.114 − 0.161i)3-s + (−3.41 + 0.658i)5-s + (1.52 − 2.16i)7-s + (0.968 + 2.79i)9-s + (−1.73 + 1.36i)11-s + (4.39 − 2.82i)13-s + (−0.286 + 0.626i)15-s + (3.43 + 3.27i)17-s + (−1.39 + 1.33i)19-s + (−0.174 − 0.494i)21-s + (0.338 + 4.78i)23-s + (6.58 − 2.63i)25-s + (1.13 + 0.332i)27-s + (4.02 − 1.18i)29-s + (7.63 − 0.728i)31-s + ⋯ |
| L(s) = 1 | + (0.0663 − 0.0931i)3-s + (−1.52 + 0.294i)5-s + (0.575 − 0.817i)7-s + (0.322 + 0.932i)9-s + (−0.523 + 0.411i)11-s + (1.21 − 0.783i)13-s + (−0.0739 + 0.161i)15-s + (0.834 + 0.795i)17-s + (−0.320 + 0.305i)19-s + (−0.0379 − 0.107i)21-s + (0.0704 + 0.997i)23-s + (1.31 − 0.527i)25-s + (0.218 + 0.0640i)27-s + (0.746 − 0.219i)29-s + (1.37 − 0.130i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21213 + 0.346448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21213 + 0.346448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.52 + 2.16i)T \) |
| 23 | \( 1 + (-0.338 - 4.78i)T \) |
| good | 3 | \( 1 + (-0.114 + 0.161i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (3.41 - 0.658i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (1.73 - 1.36i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-4.39 + 2.82i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.43 - 3.27i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.39 - 1.33i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-4.02 + 1.18i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-7.63 + 0.728i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-3.15 - 9.10i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-1.87 - 2.16i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.251 + 0.551i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-1.93 + 3.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.361 - 7.58i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.97 + 1.01i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (1.07 + 1.51i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (10.5 - 4.23i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.551 + 3.83i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.779 + 3.21i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.454 + 9.54i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-7.93 + 9.15i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.27 - 0.407i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-5.65 - 6.52i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.53452688556129729390534926724, −10.26488657293468367478211212023, −8.439166673551939388833530061325, −7.82915828318662575166635543462, −7.59373813440688506274830714441, −6.25640199749846637567751281786, −4.88093357451026856158666040427, −4.07950676985704901312614807248, −3.11962918596893638590063383060, −1.24737132708460677995418263167,
0.856923378956752023292650059007, 2.86024512787941240673956719480, 3.96376472741029571028086912886, 4.73495792870966584894102043805, 6.00122161355278654524502376027, 7.01724622468783039170836490877, 8.114731180341581629958014659285, 8.584896460764580757296030954117, 9.384084047648292702017228491770, 10.74225069308050040088998000648
