| L(s) = 1 | + 1.68i·3-s − 5-s + (0.695 + 2.55i)7-s + 0.163·9-s − 1.45·11-s − 5.12·13-s − 1.68i·15-s + 0.313i·17-s − 0.250i·19-s + (−4.29 + 1.17i)21-s + 4.27i·23-s + 25-s + 5.32i·27-s + 1.63i·29-s − 8.96·31-s + ⋯ |
| L(s) = 1 | + 0.972i·3-s − 0.447·5-s + (0.262 + 0.964i)7-s + 0.0544·9-s − 0.438·11-s − 1.42·13-s − 0.434i·15-s + 0.0761i·17-s − 0.0573i·19-s + (−0.938 + 0.255i)21-s + 0.890i·23-s + 0.200·25-s + 1.02i·27-s + 0.304i·29-s − 1.60·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7634117634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7634117634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.695 - 2.55i)T \) |
| good | 3 | \( 1 - 1.68iT - 3T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 0.313iT - 17T^{2} \) |
| 19 | \( 1 + 0.250iT - 19T^{2} \) |
| 23 | \( 1 - 4.27iT - 23T^{2} \) |
| 29 | \( 1 - 1.63iT - 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 3.47iT - 37T^{2} \) |
| 41 | \( 1 + 9.88iT - 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 + 1.90iT - 53T^{2} \) |
| 59 | \( 1 - 7.73iT - 59T^{2} \) |
| 61 | \( 1 + 0.415T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 1.50iT - 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 + 9.36iT - 79T^{2} \) |
| 83 | \( 1 - 3.45iT - 83T^{2} \) |
| 89 | \( 1 + 9.12iT - 89T^{2} \) |
| 97 | \( 1 - 16.5iT - 97T^{2} \) |
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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
−10.24602186432011735748331264753, −9.324531036491360325492162796624, −8.941576270959454305055118125808, −7.70313881170153166279497127384, −7.17771049959184793102020378924, −5.66087622833937345249912041230, −5.11371620907826497278605462188, −4.23027880710070990734250169938, −3.20091159834734185147394181601, −2.04439432169986665036911698767,
0.32147641455116785659640311840, 1.70343868084902920611587348328, 2.89517896239311793084606110271, 4.24544410937891981968628958840, 4.95121848041511297562611503544, 6.28748639444029922240251308948, 7.16596116463029715765798554208, 7.58963835843630498660440297355, 8.233850377574953224887708059281, 9.500894341342707972570206882980
