| L(s) = 1 | + 2.37·2-s + 3.63·4-s + 3.84·5-s − 7-s + 3.87·8-s + 9.11·10-s + 3.34·11-s + 4.24·13-s − 2.37·14-s + 1.92·16-s − 3.68·17-s − 1.16·19-s + 13.9·20-s + 7.93·22-s + 1.32·23-s + 9.75·25-s + 10.0·26-s − 3.63·28-s − 6.59·29-s − 3.12·31-s − 3.17·32-s − 8.74·34-s − 3.84·35-s + 5.94·37-s − 2.75·38-s + 14.8·40-s + 10.3·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.81·4-s + 1.71·5-s − 0.377·7-s + 1.36·8-s + 2.88·10-s + 1.00·11-s + 1.17·13-s − 0.634·14-s + 0.480·16-s − 0.893·17-s − 0.266·19-s + 3.11·20-s + 1.69·22-s + 0.276·23-s + 1.95·25-s + 1.97·26-s − 0.686·28-s − 1.22·29-s − 0.560·31-s − 0.561·32-s − 1.49·34-s − 0.649·35-s + 0.977·37-s − 0.447·38-s + 2.35·40-s + 1.62·41-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
| \(L(1)\) |
\(\approx\) |
\(8.728452075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.728452075\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.75T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 - 3.62T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 + 1.82T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−7.42542888880929693124402652458, −6.55788928111429745829948822541, −6.30202391188745074956332303408, −5.77506077852098240904319499611, −5.16796472452983827440435100860, −4.19557933620811567240161541098, −3.74097054513292095311381188717, −2.72711544125108791955837836397, −2.10485818911751468724341225974, −1.26119719122425144761175459849,
1.26119719122425144761175459849, 2.10485818911751468724341225974, 2.72711544125108791955837836397, 3.74097054513292095311381188717, 4.19557933620811567240161541098, 5.16796472452983827440435100860, 5.77506077852098240904319499611, 6.30202391188745074956332303408, 6.55788928111429745829948822541, 7.42542888880929693124402652458
