| L(s) = 1 | + (0.978 + 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (1.63 + 1.81i)5-s + (0.309 − 0.951i)6-s + (2.49 + 0.893i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (1.22 + 2.11i)10-s + (−2.60 + 2.05i)11-s + (0.5 − 0.866i)12-s + (−0.137 − 0.422i)13-s + (2.25 + 1.39i)14-s + (1.97 − 1.43i)15-s + (0.669 + 0.743i)16-s + (−3.96 + 0.843i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (0.0603 − 0.574i)3-s + (0.456 + 0.203i)4-s + (0.731 + 0.812i)5-s + (0.126 − 0.388i)6-s + (0.941 + 0.337i)7-s + (0.286 + 0.207i)8-s + (−0.326 − 0.0693i)9-s + (0.386 + 0.669i)10-s + (−0.784 + 0.620i)11-s + (0.144 − 0.250i)12-s + (−0.0380 − 0.117i)13-s + (0.601 + 0.371i)14-s + (0.510 − 0.370i)15-s + (0.167 + 0.185i)16-s + (−0.962 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42108 + 0.418664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42108 + 0.418664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 7 | \( 1 + (-2.49 - 0.893i)T \) |
| 11 | \( 1 + (2.60 - 2.05i)T \) |
| good | 5 | \( 1 + (-1.63 - 1.81i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (0.137 + 0.422i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.96 - 0.843i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.469i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-3.35 + 5.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.24 + 0.905i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.00 - 1.11i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.803 + 7.64i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (4.87 + 3.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + (-2.47 + 1.10i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (6.59 - 7.32i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (3.67 + 1.63i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (10.2 + 11.3i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.131 + 0.227i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 4.72i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.14 + 1.40i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.732 - 0.155i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (5.43 - 16.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (7.45 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.89 + 8.92i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.90781492918687813291601434613, −10.69548388123170271371124899623, −9.235666056198538087459035148913, −8.147887591718138010723519929998, −7.23831344678451304079635006966, −6.43717595118606844279802549868, −5.45947000394280741426984339711, −4.49767326666406261976290780231, −2.74030154778430627848099775507, −2.03322071046324831121952349421,
1.53766192100600300242181956105, 3.02085709670779245129661591685, 4.45494704330470691254607817847, 5.11165830439396357178585350957, 5.87900620539564810009329849039, 7.31653292001867288505139405712, 8.414382349772701963373172215052, 9.251072654845397552735006957634, 10.24850207965396911645273179470, 11.10145289394167525113434267031
