| L(s) = 1 | − 2.49·2-s + 4.22·4-s − 0.373·5-s − 7-s − 5.54·8-s + 0.932·10-s + 6.40·11-s + 2.54·13-s + 2.49·14-s + 5.38·16-s − 4.88·17-s − 5.83·19-s − 1.57·20-s − 15.9·22-s + 8.35·23-s − 4.86·25-s − 6.34·26-s − 4.22·28-s + 1.63·29-s − 6.96·31-s − 2.35·32-s + 12.1·34-s + 0.373·35-s + 7.46·37-s + 14.5·38-s + 2.07·40-s + 9.02·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.11·4-s − 0.167·5-s − 0.377·7-s − 1.96·8-s + 0.294·10-s + 1.93·11-s + 0.705·13-s + 0.666·14-s + 1.34·16-s − 1.18·17-s − 1.33·19-s − 0.352·20-s − 3.40·22-s + 1.74·23-s − 0.972·25-s − 1.24·26-s − 0.798·28-s + 0.303·29-s − 1.25·31-s − 0.415·32-s + 2.09·34-s + 0.0631·35-s + 1.22·37-s + 2.36·38-s + 0.327·40-s + 1.40·41-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 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 + 0.373T + 5T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 + 6.14T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 7.04T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 + 7.21T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.59812970908898895485377680528, −6.71541692473629138659909450092, −6.63203292893840354455706428314, −5.87208673809112509434680885907, −4.44393883238393125821278735787, −3.83539322885809506495770430486, −2.79186717337905710957243148017, −1.81790832717354732237654157519, −1.13223073178406970027683502675, 0,
1.13223073178406970027683502675, 1.81790832717354732237654157519, 2.79186717337905710957243148017, 3.83539322885809506495770430486, 4.44393883238393125821278735787, 5.87208673809112509434680885907, 6.63203292893840354455706428314, 6.71541692473629138659909450092, 7.59812970908898895485377680528
