| L(s) = 1 | + 0.415·2-s − 1.82·4-s + 0.233·5-s − 7-s − 1.59·8-s + 0.0972·10-s − 2.55·11-s + 1.50·13-s − 0.415·14-s + 2.99·16-s − 0.262·17-s + 0.541·19-s − 0.427·20-s − 1.06·22-s + 3.11·23-s − 4.94·25-s + 0.627·26-s + 1.82·28-s + 2.23·29-s + 8.80·31-s + 4.42·32-s − 0.109·34-s − 0.233·35-s − 9.13·37-s + 0.225·38-s − 0.372·40-s + 7.98·41-s + ⋯ |
| L(s) = 1 | + 0.294·2-s − 0.913·4-s + 0.104·5-s − 0.377·7-s − 0.562·8-s + 0.0307·10-s − 0.770·11-s + 0.418·13-s − 0.111·14-s + 0.748·16-s − 0.0636·17-s + 0.124·19-s − 0.0955·20-s − 0.226·22-s + 0.650·23-s − 0.989·25-s + 0.123·26-s + 0.345·28-s + 0.414·29-s + 1.58·31-s + 0.782·32-s − 0.0187·34-s − 0.0395·35-s − 1.50·37-s + 0.0365·38-s − 0.0588·40-s + 1.24·41-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 | \( 1 - 0.415T + 2T^{2} \) |
| 5 | \( 1 - 0.233T + 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 0.262T + 17T^{2} \) |
| 19 | \( 1 - 0.541T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 + 9.13T + 37T^{2} \) |
| 41 | \( 1 - 7.98T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 0.296T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 9.14T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.69320590577127883248505636712, −6.56931191803659984508683761244, −6.11970706599672712368948083821, −5.18884288756874986019231373599, −4.82832755554918666701935879839, −3.86853998060277159400601222846, −3.25484778840063260623141337713, −2.40825969873832831714321913177, −1.10685461225756400494612377115, 0,
1.10685461225756400494612377115, 2.40825969873832831714321913177, 3.25484778840063260623141337713, 3.86853998060277159400601222846, 4.82832755554918666701935879839, 5.18884288756874986019231373599, 6.11970706599672712368948083821, 6.56931191803659984508683761244, 7.69320590577127883248505636712
