L(s) = 1 | + (−1.60 − 2.77i)5-s + (1.02 − 2.43i)7-s + (−2.12 + 3.68i)11-s − 3.15·13-s + (−2.20 + 3.81i)17-s + (1.57 + 2.73i)19-s + (−2.20 − 3.81i)23-s + (−2.62 + 4.54i)25-s − 7.20·29-s + (1.02 − 1.77i)31-s + (−8.40 + 1.06i)35-s + (4.82 + 8.35i)37-s + 10.4·41-s − 0.750·43-s + (1.20 + 2.08i)47-s + ⋯ |
L(s) = 1 | + (−0.715 − 1.23i)5-s + (0.387 − 0.922i)7-s + (−0.640 + 1.10i)11-s − 0.874·13-s + (−0.533 + 0.924i)17-s + (0.361 + 0.626i)19-s + (−0.459 − 0.795i)23-s + (−0.524 + 0.909i)25-s − 1.33·29-s + (0.183 − 0.318i)31-s + (−1.42 + 0.180i)35-s + (0.793 + 1.37i)37-s + 1.63·41-s − 0.114·43-s + (0.175 + 0.303i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4703050164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4703050164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.02 + 2.43i)T \) |
good | 5 | \( 1 + (1.60 + 2.77i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.12 - 3.68i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + (2.20 - 3.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.57 - 2.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.20 + 3.81i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.20T + 29T^{2} \) |
| 31 | \( 1 + (-1.02 + 1.77i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.82 - 8.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 0.750T + 43T^{2} \) |
| 47 | \( 1 + (-1.20 - 2.08i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.64 - 2.85i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.12 - 7.14i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.04 - 7.01i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.40T + 71T^{2} \) |
| 73 | \( 1 + (7.62 - 13.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.22 + 14.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + (-1.20 - 2.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−9.367641917032093925155793660624, −8.409016698036941445886031786065, −7.72524267497233570147732168730, −7.39739234077632379070800083885, −6.15796283474410126341925196037, −5.05097160679363549016898403524, −4.43991440117219630056343807179, −3.95189276788748560077283383409, −2.35169671312759799069806722249, −1.19830354541306160240875287771,
0.17682678000027265359468528391, 2.31190695290768518627413172043, 2.88650981676102058018954180813, 3.81944258249612148177721423255, 5.04653300330250447935068583154, 5.71015440908578022739075350467, 6.66814425730082856860868795993, 7.58609245794673235087246367398, 7.86486442866553056859019834430, 9.052642039372043176652044217441