| L(s) = 1 | + 9.03·2-s + 9·3-s + 49.5·4-s + 81.2·6-s + 36.7·7-s + 158.·8-s + 81·9-s − 121·11-s + 446.·12-s − 982.·13-s + 331.·14-s − 151.·16-s − 870.·17-s + 731.·18-s − 2.03e3·19-s + 330.·21-s − 1.09e3·22-s + 132.·23-s + 1.43e3·24-s − 8.87e3·26-s + 729·27-s + 1.82e3·28-s + 2.73e3·29-s − 1.40e3·31-s − 6.45e3·32-s − 1.08e3·33-s − 7.86e3·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.54·4-s + 0.921·6-s + 0.283·7-s + 0.877·8-s + 0.333·9-s − 0.301·11-s + 0.894·12-s − 1.61·13-s + 0.452·14-s − 0.148·16-s − 0.730·17-s + 0.532·18-s − 1.29·19-s + 0.163·21-s − 0.481·22-s + 0.0521·23-s + 0.506·24-s − 2.57·26-s + 0.192·27-s + 0.439·28-s + 0.603·29-s − 0.262·31-s − 1.11·32-s − 0.174·33-s − 1.16·34-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 | \( 1 - 9.03T + 32T^{2} \) |
| 7 | \( 1 - 36.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 982.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 870.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 132.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 693.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.27e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.72e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.47e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.25e4T + 8.58e9T^{2} \) |
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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
−8.990352265661976199851261158347, −8.046922167154735306897604705937, −7.04277644565035205626081036827, −6.37966357553941842311673306237, −5.09871894470142268435638650843, −4.65380959165943995805673755275, −3.67247399911004841120126356402, −2.59627031508510415099300795652, −2.00400854961734466567857302850, 0,
2.00400854961734466567857302850, 2.59627031508510415099300795652, 3.67247399911004841120126356402, 4.65380959165943995805673755275, 5.09871894470142268435638650843, 6.37966357553941842311673306237, 7.04277644565035205626081036827, 8.046922167154735306897604705937, 8.990352265661976199851261158347
