| L(s) = 1 | + 2.71·2-s + 5.37·4-s − 1.33·5-s − 7-s + 9.15·8-s − 3.63·10-s − 5.10·11-s − 0.127·13-s − 2.71·14-s + 14.1·16-s − 3.48·17-s + 0.837·19-s − 7.18·20-s − 13.8·22-s − 5.34·23-s − 3.20·25-s − 0.346·26-s − 5.37·28-s + 2.80·29-s − 4.97·31-s + 19.9·32-s − 9.45·34-s + 1.33·35-s − 5.80·37-s + 2.27·38-s − 12.2·40-s − 0.0808·41-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.68·4-s − 0.598·5-s − 0.377·7-s + 3.23·8-s − 1.14·10-s − 1.54·11-s − 0.0354·13-s − 0.725·14-s + 3.52·16-s − 0.844·17-s + 0.192·19-s − 1.60·20-s − 2.95·22-s − 1.11·23-s − 0.641·25-s − 0.0680·26-s − 1.01·28-s + 0.520·29-s − 0.893·31-s + 3.53·32-s − 1.62·34-s + 0.226·35-s − 0.953·37-s + 0.369·38-s − 1.93·40-s − 0.0126·41-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 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 0.127T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 - 0.837T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 0.0808T + 41T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 0.340T + 97T^{2} \) |
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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
−7.20136306196525194617247803509, −6.69156109218873783461025483648, −5.82617353402246504378753117034, −5.34093537437000343131959687782, −4.64084688921109130766968364067, −3.94139678639982804242493924861, −3.32747225265940062956912333964, −2.54369660262962448700396679112, −1.86286201576710699103293567569, 0,
1.86286201576710699103293567569, 2.54369660262962448700396679112, 3.32747225265940062956912333964, 3.94139678639982804242493924861, 4.64084688921109130766968364067, 5.34093537437000343131959687782, 5.82617353402246504378753117034, 6.69156109218873783461025483648, 7.20136306196525194617247803509
