| L(s) = 1 | + 1.60·2-s + 0.574·4-s − 0.248·5-s + 7-s − 2.28·8-s − 0.398·10-s − 2.06·11-s + 4.61·13-s + 1.60·14-s − 4.81·16-s + 1.38·17-s − 2.86·19-s − 0.142·20-s − 3.31·22-s − 5.17·23-s − 4.93·25-s + 7.40·26-s + 0.574·28-s + 9.35·29-s − 7.06·31-s − 3.15·32-s + 2.22·34-s − 0.248·35-s + 9.60·37-s − 4.59·38-s + 0.568·40-s − 7.82·41-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.287·4-s − 0.111·5-s + 0.377·7-s − 0.808·8-s − 0.126·10-s − 0.622·11-s + 1.27·13-s + 0.428·14-s − 1.20·16-s + 0.335·17-s − 0.657·19-s − 0.0319·20-s − 0.706·22-s − 1.07·23-s − 0.987·25-s + 1.45·26-s + 0.108·28-s + 1.73·29-s − 1.26·31-s − 0.558·32-s + 0.381·34-s − 0.0420·35-s + 1.57·37-s − 0.745·38-s + 0.0898·40-s − 1.22·41-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 | \( 1 - 1.60T + 2T^{2} \) |
| 5 | \( 1 + 0.248T + 5T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 9.35T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 9.83T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 5.88T + 97T^{2} \) |
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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
−7.54155557723032522786285029214, −6.41239683617906236462951400672, −5.96166119222594486085029656361, −5.46115878432160774752109048327, −4.37840951638794968056055473616, −4.20704164182581794187320850060, −3.25089615377244209499084577682, −2.52469099461902483344739570387, −1.43668394921816853478810816779, 0,
1.43668394921816853478810816779, 2.52469099461902483344739570387, 3.25089615377244209499084577682, 4.20704164182581794187320850060, 4.37840951638794968056055473616, 5.46115878432160774752109048327, 5.96166119222594486085029656361, 6.41239683617906236462951400672, 7.54155557723032522786285029214
