| L(s) = 1 | + 1.68·2-s + 2.23·3-s + 0.833·4-s + 3.26·5-s + 3.76·6-s + 1.22·7-s − 1.96·8-s + 2.01·9-s + 5.49·10-s + 11-s + 1.86·12-s − 1.47·13-s + 2.05·14-s + 7.30·15-s − 4.97·16-s + 3.68·17-s + 3.38·18-s − 4.90·19-s + 2.71·20-s + 2.73·21-s + 1.68·22-s − 3.22·23-s − 4.39·24-s + 5.64·25-s − 2.49·26-s − 2.21·27-s + 1.01·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 1.29·3-s + 0.416·4-s + 1.45·5-s + 1.53·6-s + 0.461·7-s − 0.694·8-s + 0.670·9-s + 1.73·10-s + 0.301·11-s + 0.538·12-s − 0.410·13-s + 0.548·14-s + 1.88·15-s − 1.24·16-s + 0.894·17-s + 0.797·18-s − 1.12·19-s + 0.608·20-s + 0.596·21-s + 0.358·22-s − 0.672·23-s − 0.897·24-s + 1.12·25-s − 0.488·26-s − 0.426·27-s + 0.192·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.508736688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.508736688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 - 0.352T + 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 + 7.01T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 8.31T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 - 5.61T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−9.674813812691910568095678185779, −8.631558729083089935601533185890, −8.176712426402735990235286943099, −6.83451576651264688431613904398, −6.07895711323761806676364510251, −5.27172228277936814671860890244, −4.42772728873445686293469847762, −3.41200858207258419613238199747, −2.55386563360143801976083631151, −1.78735373050874830777510308530,
1.78735373050874830777510308530, 2.55386563360143801976083631151, 3.41200858207258419613238199747, 4.42772728873445686293469847762, 5.27172228277936814671860890244, 6.07895711323761806676364510251, 6.83451576651264688431613904398, 8.176712426402735990235286943099, 8.631558729083089935601533185890, 9.674813812691910568095678185779
