| L(s) = 1 | + 4.80·2-s + 15.1·4-s + 1.59·5-s − 34.5·7-s + 34.2·8-s + 7.64·10-s + 42.6·11-s − 166.·14-s + 43.7·16-s + 45.4·17-s − 41.0·19-s + 24.0·20-s + 205.·22-s − 171.·23-s − 122.·25-s − 522.·28-s − 178.·29-s − 18.5·31-s − 63.8·32-s + 218.·34-s − 54.9·35-s + 331.·37-s − 197.·38-s + 54.4·40-s − 411.·41-s + 11.3·43-s + 645.·44-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.89·4-s + 0.142·5-s − 1.86·7-s + 1.51·8-s + 0.241·10-s + 1.16·11-s − 3.16·14-s + 0.682·16-s + 0.648·17-s − 0.496·19-s + 0.268·20-s + 1.98·22-s − 1.55·23-s − 0.979·25-s − 3.52·28-s − 1.14·29-s − 0.107·31-s − 0.352·32-s + 1.10·34-s − 0.265·35-s + 1.47·37-s − 0.843·38-s + 0.215·40-s − 1.56·41-s + 0.0402·43-s + 2.21·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.80T + 8T^{2} \) |
| 5 | \( 1 - 1.59T + 125T^{2} \) |
| 7 | \( 1 + 34.5T + 343T^{2} \) |
| 11 | \( 1 - 42.6T + 1.33e3T^{2} \) |
| 17 | \( 1 - 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 11.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 494.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 217.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 775.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 676.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 645.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 778.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 160.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 7.23T + 9.12e5T^{2} \) |
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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
−8.782869390755675852427969011699, −7.45938396065362705511209463455, −6.62300722581827815559668757059, −6.10768498895870684918655951286, −5.57530727111557369926519086515, −4.13889056564756778833105613478, −3.76245720579868536677452138209, −2.92882495162933473842987790408, −1.81906796135614986218457452066, 0,
1.81906796135614986218457452066, 2.92882495162933473842987790408, 3.76245720579868536677452138209, 4.13889056564756778833105613478, 5.57530727111557369926519086515, 6.10768498895870684918655951286, 6.62300722581827815559668757059, 7.45938396065362705511209463455, 8.782869390755675852427969011699
