| L(s) = 1 | − 1.22·2-s − 6.50·4-s + 11.7·5-s − 7·7-s + 17.7·8-s − 14.3·10-s + 11·11-s − 25.5·13-s + 8.56·14-s + 30.3·16-s − 16.3·17-s − 126.·19-s − 76.1·20-s − 13.4·22-s + 94.6·23-s + 12.2·25-s + 31.2·26-s + 45.5·28-s + 160.·29-s − 12.1·31-s − 179.·32-s + 19.9·34-s − 82.0·35-s + 436.·37-s + 154.·38-s + 207.·40-s − 264.·41-s + ⋯ |
| L(s) = 1 | − 0.432·2-s − 0.812·4-s + 1.04·5-s − 0.377·7-s + 0.784·8-s − 0.453·10-s + 0.301·11-s − 0.545·13-s + 0.163·14-s + 0.473·16-s − 0.232·17-s − 1.52·19-s − 0.851·20-s − 0.130·22-s + 0.858·23-s + 0.0979·25-s + 0.235·26-s + 0.307·28-s + 1.02·29-s − 0.0701·31-s − 0.989·32-s + 0.100·34-s − 0.396·35-s + 1.94·37-s + 0.658·38-s + 0.821·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 1.22T + 8T^{2} \) |
| 5 | \( 1 - 11.7T + 125T^{2} \) |
| 13 | \( 1 + 25.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 607.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 783.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 489.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 466.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 429.T + 9.12e5T^{2} \) |
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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
−9.710544162767554588522755751896, −8.878194449188127204568215175168, −8.182434701725897214312592908330, −6.91397090733053461628800919437, −6.10929774538131927725444006893, −5.03141479785759324148363275160, −4.17169640590165057727629389742, −2.69563789644643788545369292425, −1.43396746574585920656150309750, 0,
1.43396746574585920656150309750, 2.69563789644643788545369292425, 4.17169640590165057727629389742, 5.03141479785759324148363275160, 6.10929774538131927725444006893, 6.91397090733053461628800919437, 8.182434701725897214312592908330, 8.878194449188127204568215175168, 9.710544162767554588522755751896
