| L(s) = 1 | + (0.359 + 0.359i)3-s + (1.39 + 1.75i)5-s + (1.87 − 1.87i)7-s − 2.74i·9-s + (4.48 + 4.48i)13-s + (−0.129 + 1.12i)15-s − 7.62i·19-s + 1.34·21-s + (−0.741 − 0.741i)23-s + (−1.12 + 4.87i)25-s + (2.06 − 2.06i)27-s + (5.87 + 0.672i)35-s + 3.22i·39-s + (4.79 − 3.81i)45-s − 7i·49-s + ⋯ |
| L(s) = 1 | + (0.207 + 0.207i)3-s + (0.622 + 0.782i)5-s + (0.707 − 0.707i)7-s − 0.913i·9-s + (1.24 + 1.24i)13-s + (−0.0333 + 0.291i)15-s − 1.75i·19-s + 0.293·21-s + (−0.154 − 0.154i)23-s + (−0.225 + 0.974i)25-s + (0.397 − 0.397i)27-s + (0.993 + 0.113i)35-s + 0.516i·39-s + (0.715 − 0.568i)45-s − i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224792621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224792621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.39 - 1.75i)T \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
| good | 3 | \( 1 + (-0.359 - 0.359i)T + 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.48 - 4.48i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 7.62iT - 19T^{2} \) |
| 23 | \( 1 + (0.741 + 0.741i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 15.3iT - 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 15.7iT - 79T^{2} \) |
| 83 | \( 1 + (-5.83 - 5.83i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−9.777255264228523048579472973712, −9.092914470300167920538652947496, −8.382323667452368408481924147039, −7.00364496636285010039338410162, −6.75575879492658969568959782509, −5.70227383808296096213177663024, −4.44295782819220690110584654235, −3.70802921825022489327391669438, −2.52703751834728211075253535942, −1.21621844897670980363648807818,
1.33923628580474456271374830985, 2.19227867426395781809993512840, 3.55225653211854200401294995374, 4.84977277056708972952802202975, 5.58477781363733559716979076837, 6.13064082977682615789632386005, 7.71282727612062779615056188828, 8.256884373517847415122773057450, 8.718002412641446519096677689978, 9.859276268846991430175723094075
