| L(s) = 1 | + 2.23·2-s − 3·3-s − 3.02·4-s − 16.4·5-s − 6.69·6-s + 34.4·7-s − 24.5·8-s + 9·9-s − 36.6·10-s + 0.334·11-s + 9.07·12-s + 31.4·13-s + 76.9·14-s + 49.3·15-s − 30.6·16-s + 77.1·17-s + 20.0·18-s + 36.8·19-s + 49.7·20-s − 103.·21-s + 0.745·22-s − 100.·23-s + 73.7·24-s + 145.·25-s + 70.1·26-s − 27·27-s − 104.·28-s + ⋯ |
| L(s) = 1 | + 0.788·2-s − 0.577·3-s − 0.378·4-s − 1.47·5-s − 0.455·6-s + 1.86·7-s − 1.08·8-s + 0.333·9-s − 1.15·10-s + 0.00916·11-s + 0.218·12-s + 0.671·13-s + 1.46·14-s + 0.849·15-s − 0.478·16-s + 1.10·17-s + 0.262·18-s + 0.444·19-s + 0.555·20-s − 1.07·21-s + 0.00722·22-s − 0.914·23-s + 0.627·24-s + 1.16·25-s + 0.529·26-s − 0.192·27-s − 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 211 | \( 1 - 211T \) |
| good | 2 | \( 1 - 2.23T + 8T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 7 | \( 1 - 34.4T + 343T^{2} \) |
| 11 | \( 1 - 0.334T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 395.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 51.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 605.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 16.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 807.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 704.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 208.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 534.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 641.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 941.T + 9.12e5T^{2} \) |
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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
−9.883450675275006303892592877151, −8.497671565872519718296544798038, −8.060207984011620280674854201125, −7.16943568959127336823184119058, −5.65042662941280065426848434186, −5.08009794464215441683880145277, −4.13955240473672074518305621997, −3.52888206754385640840882137408, −1.45178067980667361026043277596, 0,
1.45178067980667361026043277596, 3.52888206754385640840882137408, 4.13955240473672074518305621997, 5.08009794464215441683880145277, 5.65042662941280065426848434186, 7.16943568959127336823184119058, 8.060207984011620280674854201125, 8.497671565872519718296544798038, 9.883450675275006303892592877151
