| L(s) = 1 | − 2.23i·2-s − 3.23i·3-s − 3.00·4-s − 7.23·6-s + i·7-s + 2.23i·8-s − 7.47·9-s − 11-s + 9.70i·12-s + 1.23i·13-s + 2.23·14-s − 0.999·16-s + 1.23i·17-s + 16.7i·18-s + 2.47·19-s + ⋯ |
| L(s) = 1 | − 1.58i·2-s − 1.86i·3-s − 1.50·4-s − 2.95·6-s + 0.377i·7-s + 0.790i·8-s − 2.49·9-s − 0.301·11-s + 2.80i·12-s + 0.342i·13-s + 0.597·14-s − 0.249·16-s + 0.299i·17-s + 3.93i·18-s + 0.567·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23iT - 2T^{2} \) |
| 3 | \( 1 + 3.23iT - 3T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 1.23iT - 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.47iT - 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472iT - 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 7.23iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.52iT - 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 5.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41iT - 97T^{2} \) |
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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
−8.517733662060441569167666292594, −7.56103458599808877805934280810, −6.96041686730827269709297753496, −5.92458493948271036092023642719, −5.14474789118311558517223263504, −3.65302736768024494399501728975, −2.86209934492742746852049379343, −1.93003945311510567224958492208, −1.36980812974648068287476140684, 0,
2.80975984627373197254422221681, 3.84029485397703723662128444184, 4.65483778642871763727819634016, 5.20464329098646842902802047754, 5.89833293066929782967808640342, 6.81613191846677392335290204561, 7.79460264439017556450710988088, 8.464907960158413596422878387958, 9.162343255595361909697953270094
