| L(s) = 1 | − 1.45·2-s + 0.117·4-s + 0.420·5-s + 7-s + 2.73·8-s − 0.611·10-s − 4.09·11-s + 5.95·13-s − 1.45·14-s − 4.22·16-s + 7.99·17-s + 1.20·19-s + 0.0491·20-s + 5.95·22-s + 0.806·23-s − 4.82·25-s − 8.66·26-s + 0.117·28-s − 5.46·29-s + 3.58·31-s + 0.661·32-s − 11.6·34-s + 0.420·35-s − 3.79·37-s − 1.74·38-s + 1.15·40-s − 9.24·41-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0585·4-s + 0.187·5-s + 0.377·7-s + 0.968·8-s − 0.193·10-s − 1.23·11-s + 1.65·13-s − 0.388·14-s − 1.05·16-s + 1.93·17-s + 0.275·19-s + 0.0109·20-s + 1.26·22-s + 0.168·23-s − 0.964·25-s − 1.70·26-s + 0.0221·28-s − 1.01·29-s + 0.644·31-s + 0.116·32-s − 1.99·34-s + 0.0709·35-s − 0.624·37-s − 0.283·38-s + 0.181·40-s − 1.44·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 - 0.420T + 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 7.99T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 + 9.24T + 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 - 8.96T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−7.79365679320016009508859904653, −7.10057754526753849229089046283, −6.06532484791985065350593650479, −5.42986897838059700889658113270, −4.84114732436266250583224712862, −3.73235321795861931113203605339, −3.13293191972145674260065180862, −1.77340315613627045001134263824, −1.24908626536862198568871970716, 0,
1.24908626536862198568871970716, 1.77340315613627045001134263824, 3.13293191972145674260065180862, 3.73235321795861931113203605339, 4.84114732436266250583224712862, 5.42986897838059700889658113270, 6.06532484791985065350593650479, 7.10057754526753849229089046283, 7.79365679320016009508859904653
