| L(s) = 1 | − 2.53·2-s − 2.04·3-s + 4.42·4-s − 2.47·5-s + 5.17·6-s − 0.629·7-s − 6.13·8-s + 1.16·9-s + 6.28·10-s + 0.0387·11-s − 9.02·12-s − 2.09·13-s + 1.59·14-s + 5.05·15-s + 6.71·16-s + 1.11·17-s − 2.94·18-s − 4.51·19-s − 10.9·20-s + 1.28·21-s − 0.0981·22-s + 4.29·23-s + 12.5·24-s + 1.14·25-s + 5.30·26-s + 3.74·27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.17·3-s + 2.21·4-s − 1.10·5-s + 2.11·6-s − 0.238·7-s − 2.17·8-s + 0.387·9-s + 1.98·10-s + 0.0116·11-s − 2.60·12-s − 0.580·13-s + 0.426·14-s + 1.30·15-s + 1.67·16-s + 0.270·17-s − 0.694·18-s − 1.03·19-s − 2.45·20-s + 0.280·21-s − 0.0209·22-s + 0.896·23-s + 2.55·24-s + 0.229·25-s + 1.04·26-s + 0.721·27-s − 0.526·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 + 0.629T + 7T^{2} \) |
| 11 | \( 1 - 0.0387T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 - 0.632T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 0.733T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 5.00T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 - 1.15T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 - 7.30T + 83T^{2} \) |
| 89 | \( 1 + 0.395T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.122722907988116976454485742777, −7.38183745945436344043829630938, −6.92392200647559156217017610181, −6.16436562656251279011118973256, −5.31230055352516022339107250121, −4.29642962693791968974106089636, −3.19044281214390364831976610447, −2.05347311891599718932237958575, −0.75499891513865478916362849259, 0,
0.75499891513865478916362849259, 2.05347311891599718932237958575, 3.19044281214390364831976610447, 4.29642962693791968974106089636, 5.31230055352516022339107250121, 6.16436562656251279011118973256, 6.92392200647559156217017610181, 7.38183745945436344043829630938, 8.122722907988116976454485742777
