| L(s) = 1 | − 1.42·2-s + 0.0424·4-s − 0.596·5-s − 4.06·7-s + 2.79·8-s + 0.852·10-s − 11-s − 6.34·13-s + 5.80·14-s − 4.08·16-s + 6.18·17-s + 4.48·19-s − 0.0253·20-s + 1.42·22-s + 3.88·23-s − 4.64·25-s + 9.06·26-s − 0.172·28-s − 2.01·29-s − 5.51·31-s + 0.240·32-s − 8.83·34-s + 2.42·35-s + 1.56·37-s − 6.41·38-s − 1.66·40-s − 0.599·41-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.0212·4-s − 0.266·5-s − 1.53·7-s + 0.989·8-s + 0.269·10-s − 0.301·11-s − 1.75·13-s + 1.55·14-s − 1.02·16-s + 1.49·17-s + 1.02·19-s − 0.00566·20-s + 0.304·22-s + 0.809·23-s − 0.928·25-s + 1.77·26-s − 0.0326·28-s − 0.374·29-s − 0.990·31-s + 0.0424·32-s − 1.51·34-s + 0.409·35-s + 0.256·37-s − 1.04·38-s − 0.263·40-s − 0.0936·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 + 0.596T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 6.18T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 - 0.217T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 + 6.30T + 71T^{2} \) |
| 73 | \( 1 - 3.74T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−7.48061831902330334022722955668, −7.26209569380697845014525131424, −6.22302320602880043656829041231, −5.37733917761596511133925429352, −4.79071013171374390803591423775, −3.66476196245538515938312744176, −3.11908907790008035875760268654, −2.13539781113888385323644486734, −0.842328760104501659143931188753, 0,
0.842328760104501659143931188753, 2.13539781113888385323644486734, 3.11908907790008035875760268654, 3.66476196245538515938312744176, 4.79071013171374390803591423775, 5.37733917761596511133925429352, 6.22302320602880043656829041231, 7.26209569380697845014525131424, 7.48061831902330334022722955668
