| L(s) = 1 | − i·2-s + 0.618i·3-s − 4-s + 0.618·6-s − 1.23i·7-s + i·8-s + 2.61·9-s − 3.61·11-s − 0.618i·12-s + 3.85i·13-s − 1.23·14-s + 16-s − 4.47i·17-s − 2.61i·18-s + 4.47·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.356i·3-s − 0.5·4-s + 0.252·6-s − 0.467i·7-s + 0.353i·8-s + 0.872·9-s − 1.09·11-s − 0.178i·12-s + 1.06i·13-s − 0.330·14-s + 0.250·16-s − 1.08i·17-s − 0.617i·18-s + 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.621315771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621315771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - 0.618iT - 3T^{2} \) |
| 7 | \( 1 + 1.23iT - 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.85iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 3.85iT - 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 + 0.763iT - 43T^{2} \) |
| 47 | \( 1 + 3.23iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 - 10.0iT - 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 4.09iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 5.52iT - 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 8.47iT - 97T^{2} \) |
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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
−9.354683497879366198261451171942, −8.498495796036655778233285381884, −7.44526682184386433857700145502, −6.98239999052939028684548059651, −5.64464636684864634968759883669, −4.72922754858616339451461474566, −4.20265353321453905308037079772, −3.10992792698423200568965324538, −2.12597988000683802157685011035, −0.76697107935115395931657663014,
1.03464245012918472202766131336, 2.45659085535513483458886357991, 3.57612631853737058829421293309, 4.66255204385057880488538732034, 5.59609302325021675928437883982, 6.03458464247237014846399394355, 7.25085744642685467874579396101, 7.71373726242859919279486812509, 8.308614979693931051335706055363, 9.321166840163622194469257838391
