| L(s) = 1 | − 3.24·3-s + (−1.49 − 1.65i)5-s + 7-s + 7.55·9-s − 0.886i·11-s − 0.479i·13-s + (4.87 + 5.39i)15-s + (−0.0707 + 4.12i)17-s − 3.23·19-s − 3.24·21-s − 0.245·23-s + (−0.506 + 4.97i)25-s − 14.8·27-s + 2.80i·29-s − 9.97i·31-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + (−0.670 − 0.742i)5-s + 0.377·7-s + 2.51·9-s − 0.267i·11-s − 0.132i·13-s + (1.25 + 1.39i)15-s + (−0.0171 + 0.999i)17-s − 0.742·19-s − 0.709·21-s − 0.0511·23-s + (−0.101 + 0.994i)25-s − 2.84·27-s + 0.520i·29-s − 1.79i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4941458583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4941458583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.49 + 1.65i)T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + (0.0707 - 4.12i)T \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 11 | \( 1 + 0.886iT - 11T^{2} \) |
| 13 | \( 1 + 0.479iT - 13T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 0.245T + 23T^{2} \) |
| 29 | \( 1 - 2.80iT - 29T^{2} \) |
| 31 | \( 1 + 9.97iT - 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 0.773iT - 41T^{2} \) |
| 43 | \( 1 + 0.972iT - 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 2.66iT - 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 + 12.5iT - 67T^{2} \) |
| 71 | \( 1 - 9.47iT - 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 13.0iT - 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 0.686T + 97T^{2} \) |
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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
−9.027776233544505787745181815554, −8.199525084563913491364590272007, −7.43083167109003280137250296516, −6.58068136909177960002948780521, −5.79552599736653185148399098160, −5.22972170048111677909513447410, −4.37959487585892545971480211700, −3.82243773212055388199311385744, −1.81764019696532324754654566889, −0.74000151642753262351819086353,
0.34216866878517421396281089301, 1.73724359746768941678641681152, 3.22600293057710224245831016156, 4.46212153541102130633040562135, 4.80725668699350487587005849058, 5.84057110177238979264476452180, 6.56390152865669475991024173537, 7.13402305094997321047501748405, 7.77971529130909262134935687220, 8.941779491497800682737255567810
