| L(s) = 1 | + 1.09·3-s + (2.20 + 0.370i)5-s + i·7-s − 1.80·9-s + 1.07i·11-s − 3.77·13-s + (2.41 + 0.405i)15-s + 5.08i·17-s + 0.279i·19-s + 1.09i·21-s + 4.13i·23-s + (4.72 + 1.63i)25-s − 5.25·27-s + 4.87i·29-s − 10.1·31-s + ⋯ |
| L(s) = 1 | + 0.631·3-s + (0.986 + 0.165i)5-s + 0.377i·7-s − 0.601·9-s + 0.323i·11-s − 1.04·13-s + (0.622 + 0.104i)15-s + 1.23i·17-s + 0.0641i·19-s + 0.238i·21-s + 0.861i·23-s + (0.944 + 0.327i)25-s − 1.01·27-s + 0.904i·29-s − 1.81·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0950 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0950 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.942658958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942658958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.20 - 0.370i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 11 | \( 1 - 1.07iT - 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 - 5.08iT - 17T^{2} \) |
| 19 | \( 1 - 0.279iT - 19T^{2} \) |
| 23 | \( 1 - 4.13iT - 23T^{2} \) |
| 29 | \( 1 - 4.87iT - 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 4.78iT - 47T^{2} \) |
| 53 | \( 1 - 5.30T + 53T^{2} \) |
| 59 | \( 1 - 5.22iT - 59T^{2} \) |
| 61 | \( 1 - 14.5iT - 61T^{2} \) |
| 67 | \( 1 - 9.74T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 9.92iT - 97T^{2} \) |
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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
−9.197921363734996155911436827305, −8.686704914952266729160853045183, −7.66085493905655448122370676966, −7.06428356296351773143674357962, −5.76352876800855720171567386770, −5.65584841175490602058121496937, −4.37749977669341331282427160464, −3.25824863781402301794840540161, −2.43255701216390864689596858640, −1.66315512542247440590519106721,
0.56284127767724905056019265174, 2.21789563676442247789953817169, 2.67906802232075267131792078401, 3.84410580341793147374989383421, 4.96087668681422589477857736199, 5.58404975087850475669899126155, 6.51679186517492015820021539862, 7.37301971330990946277561858570, 8.073077728913860984511591970457, 9.075876344418877664014308072947
