| L(s) = 1 | − 0.955·2-s − 1.08·4-s + 2.56·5-s − 7-s + 2.94·8-s − 2.44·10-s − 0.200·11-s + 6.35·13-s + 0.955·14-s − 0.642·16-s − 2.09·17-s − 4.55·19-s − 2.78·20-s + 0.191·22-s − 6.44·23-s + 1.57·25-s − 6.06·26-s + 1.08·28-s + 5.01·29-s + 2.16·31-s − 5.28·32-s + 2.00·34-s − 2.56·35-s + 1.92·37-s + 4.34·38-s + 7.56·40-s − 2.64·41-s + ⋯ |
| L(s) = 1 | − 0.675·2-s − 0.543·4-s + 1.14·5-s − 0.377·7-s + 1.04·8-s − 0.774·10-s − 0.0604·11-s + 1.76·13-s + 0.255·14-s − 0.160·16-s − 0.508·17-s − 1.04·19-s − 0.623·20-s + 0.0408·22-s − 1.34·23-s + 0.315·25-s − 1.18·26-s + 0.205·28-s + 0.931·29-s + 0.388·31-s − 0.934·32-s + 0.343·34-s − 0.433·35-s + 0.316·37-s + 0.705·38-s + 1.19·40-s − 0.413·41-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 | \( 1 + 0.955T + 2T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 + 0.200T + 11T^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 4.05T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 + 8.95T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 + 0.482T + 71T^{2} \) |
| 73 | \( 1 + 1.76T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.70809011269626690515258250270, −6.62928514594577153219363267571, −6.14678463287217809603691334010, −5.64984788468853456798849448465, −4.52671172824668910487408046016, −4.05722581806462156249779022698, −3.00125154478505010633367449842, −1.92167799124452002490404258866, −1.28529816939152074980905417913, 0,
1.28529816939152074980905417913, 1.92167799124452002490404258866, 3.00125154478505010633367449842, 4.05722581806462156249779022698, 4.52671172824668910487408046016, 5.64984788468853456798849448465, 6.14678463287217809603691334010, 6.62928514594577153219363267571, 7.70809011269626690515258250270
