| L(s) = 1 | + (0.978 + 0.207i)2-s + (0.0366 + 0.112i)3-s + (0.913 + 0.406i)4-s + (0.439 + 4.18i)5-s + (0.0124 + 0.118i)6-s + (−2.93 − 3.25i)7-s + (0.809 + 0.587i)8-s + (2.41 − 1.75i)9-s + (−0.439 + 4.18i)10-s − 1.98·11-s + (−0.0124 + 0.118i)12-s + (2.10 − 3.64i)13-s + (−2.19 − 3.79i)14-s + (−0.456 + 0.203i)15-s + (0.669 + 0.743i)16-s + (−0.844 − 0.376i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (0.0211 + 0.0651i)3-s + (0.456 + 0.203i)4-s + (0.196 + 1.87i)5-s + (0.00506 + 0.0481i)6-s + (−1.10 − 1.23i)7-s + (0.286 + 0.207i)8-s + (0.805 − 0.585i)9-s + (−0.139 + 1.32i)10-s − 0.598·11-s + (−0.00358 + 0.0340i)12-s + (0.584 − 1.01i)13-s + (−0.585 − 1.01i)14-s + (−0.117 + 0.0524i)15-s + (0.167 + 0.185i)16-s + (−0.204 − 0.0912i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40290 + 0.465856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40290 + 0.465856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + (-2.26 - 7.47i)T \) |
| good | 3 | \( 1 + (-0.0366 - 0.112i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.439 - 4.18i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (2.93 + 3.25i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 + (-2.10 + 3.64i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.844 + 0.376i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.37 + 2.63i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (5.72 - 4.16i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.990 - 1.71i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.234 - 0.0499i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (1.10 - 3.40i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.40 + 4.33i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.454 - 0.202i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-3.36 - 5.82i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.52 + 6.91i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.5 - 2.46i)T + (53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (-0.0734 - 0.698i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (0.505 - 4.80i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (0.00362 - 0.0344i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (3.22 - 1.43i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (9.66 + 2.05i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (2.64 + 8.14i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 2.23i)T + (88.6 - 39.4i)T^{2} \) |
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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
−13.57184463710403154203094483892, −12.91348234150610323337111588155, −11.37133226636164005443665424497, −10.34434351238811603445909390114, −9.943234662809848660662109652887, −7.52874131714333579821021985882, −6.91026988632409729578852003446, −5.97054561563895966823864510794, −3.81815283488015815501970005242, −3.04499400472559278268972800662,
2.01671101789091258871178424059, 4.15573246553654684736146109951, 5.30116979119916066510972554908, 6.27406241499117391212896167004, 8.112778051211187818091428153200, 9.160752017223335920522495054494, 10.04722929402591442082365192874, 11.80920526103168944105337068586, 12.56947410113536597278021531053, 13.05400197363255662633220103412
