| L(s) = 1 | + 0.485·2-s − 3-s − 1.76·4-s − 0.485·6-s − 1.94·7-s − 1.82·8-s + 9-s + 5.30·11-s + 1.76·12-s − 6.00·13-s − 0.946·14-s + 2.64·16-s − 1.89·17-s + 0.485·18-s − 4.84·19-s + 1.94·21-s + 2.57·22-s + 3.19·23-s + 1.82·24-s − 2.91·26-s − 27-s + 3.43·28-s + 29-s − 4.96·31-s + 4.93·32-s − 5.30·33-s − 0.919·34-s + ⋯ |
| L(s) = 1 | + 0.343·2-s − 0.577·3-s − 0.882·4-s − 0.198·6-s − 0.736·7-s − 0.646·8-s + 0.333·9-s + 1.59·11-s + 0.509·12-s − 1.66·13-s − 0.252·14-s + 0.660·16-s − 0.459·17-s + 0.114·18-s − 1.11·19-s + 0.425·21-s + 0.548·22-s + 0.666·23-s + 0.373·24-s − 0.571·26-s − 0.192·27-s + 0.649·28-s + 0.185·29-s − 0.892·31-s + 0.872·32-s − 0.923·33-s − 0.157·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8906213002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8906213002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.485T + 2T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 0.749T + 41T^{2} \) |
| 43 | \( 1 - 6.21T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.063794477476051403785574609987, −8.605399252045845084919470058412, −7.13778644819628050285745915289, −6.78137225053746667201585199838, −5.80497315401485412832152383365, −5.04146713000238500582752919981, −4.21223663932762439533517219021, −3.58670327736621200527163270118, −2.22887168477300029850118123728, −0.59146369806927440250452264530,
0.59146369806927440250452264530, 2.22887168477300029850118123728, 3.58670327736621200527163270118, 4.21223663932762439533517219021, 5.04146713000238500582752919981, 5.80497315401485412832152383365, 6.78137225053746667201585199838, 7.13778644819628050285745915289, 8.605399252045845084919470058412, 9.063794477476051403785574609987
