| L(s) = 1 | + 3.12·2-s + 1.79·4-s − 20.5·5-s + 28.8·7-s − 19.4·8-s − 64.4·10-s + 58.6·11-s + 76.6·13-s + 90.4·14-s − 75.1·16-s − 51.0·17-s − 80.5·19-s − 36.9·20-s + 183.·22-s − 7.88·23-s + 299.·25-s + 239.·26-s + 51.8·28-s − 68.8·29-s + 73.6·31-s − 79.7·32-s − 159.·34-s − 595.·35-s − 413.·37-s − 252.·38-s + 399.·40-s − 227.·41-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.224·4-s − 1.84·5-s + 1.56·7-s − 0.858·8-s − 2.03·10-s + 1.60·11-s + 1.63·13-s + 1.72·14-s − 1.17·16-s − 0.727·17-s − 0.972·19-s − 0.413·20-s + 1.77·22-s − 0.0715·23-s + 2.39·25-s + 1.80·26-s + 0.349·28-s − 0.440·29-s + 0.426·31-s − 0.440·32-s − 0.805·34-s − 2.87·35-s − 1.83·37-s − 1.07·38-s + 1.58·40-s − 0.866·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.151937565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.151937565\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 3.12T + 8T^{2} \) |
| 5 | \( 1 + 20.5T + 125T^{2} \) |
| 7 | \( 1 - 28.8T + 343T^{2} \) |
| 11 | \( 1 - 58.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.88T + 1.21e4T^{2} \) |
| 29 | \( 1 + 68.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 413.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 107.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 14.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 42.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 468.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 128.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 210.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 974.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.58e3T + 9.12e5T^{2} \) |
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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
−8.479085430492953661619422689904, −8.215178090576489811820935503614, −6.92596616550591167678881695462, −6.44023904004194754934228643020, −5.19608161441706459236118012103, −4.49871652794938861234895123130, −3.81429003792900279401644314595, −3.60280557995233875988850119908, −1.86618960118624959738058226939, −0.70438568596587855856874012923,
0.70438568596587855856874012923, 1.86618960118624959738058226939, 3.60280557995233875988850119908, 3.81429003792900279401644314595, 4.49871652794938861234895123130, 5.19608161441706459236118012103, 6.44023904004194754934228643020, 6.92596616550591167678881695462, 8.215178090576489811820935503614, 8.479085430492953661619422689904
