| L(s) = 1 | + 9.20·2-s + 9·3-s + 52.7·4-s + 82.8·6-s − 97.7·7-s + 191.·8-s + 81·9-s + 121·11-s + 474.·12-s − 490.·13-s − 900.·14-s + 72.4·16-s − 881.·17-s + 745.·18-s + 34.4·19-s − 880.·21-s + 1.11e3·22-s − 2.90e3·23-s + 1.72e3·24-s − 4.51e3·26-s + 729·27-s − 5.16e3·28-s − 1.41e3·29-s − 2.53e3·31-s − 5.45e3·32-s + 1.08e3·33-s − 8.11e3·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.64·4-s + 0.939·6-s − 0.754·7-s + 1.05·8-s + 0.333·9-s + 0.301·11-s + 0.952·12-s − 0.804·13-s − 1.22·14-s + 0.0707·16-s − 0.739·17-s + 0.542·18-s + 0.0218·19-s − 0.435·21-s + 0.490·22-s − 1.14·23-s + 0.610·24-s − 1.30·26-s + 0.192·27-s − 1.24·28-s − 0.311·29-s − 0.473·31-s − 0.941·32-s + 0.174·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 9.20T + 32T^{2} \) |
| 7 | \( 1 + 97.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 490.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 881.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 34.4T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.81e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.61e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 672.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.69e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.44e4T + 8.58e9T^{2} \) |
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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
−9.179200678741057061984169472129, −7.996844657776916793041329983296, −6.96222080138159621986326440989, −6.35792391146531416067623378292, −5.36934793290220465804183059579, −4.38674334546592457614527094584, −3.66627820621278471982509115686, −2.76579645175307241934709950351, −1.91225835782658919833327896581, 0,
1.91225835782658919833327896581, 2.76579645175307241934709950351, 3.66627820621278471982509115686, 4.38674334546592457614527094584, 5.36934793290220465804183059579, 6.35792391146531416067623378292, 6.96222080138159621986326440989, 7.996844657776916793041329983296, 9.179200678741057061984169472129
