| L(s) = 1 | − 9.15·2-s + 51.7·4-s − 95.5·5-s + 209.·7-s − 180.·8-s + 874.·10-s + 335.·13-s − 1.91e3·14-s − 1.19·16-s − 799.·17-s + 658.·19-s − 4.94e3·20-s + 4.11e3·23-s + 5.99e3·25-s − 3.07e3·26-s + 1.08e4·28-s + 559.·29-s − 6.05e3·31-s + 5.79e3·32-s + 7.31e3·34-s − 2.00e4·35-s − 1.40e4·37-s − 6.02e3·38-s + 1.72e4·40-s + 1.84e3·41-s − 1.62e3·43-s − 3.77e4·46-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s − 1.70·5-s + 1.61·7-s − 0.999·8-s + 2.76·10-s + 0.551·13-s − 2.61·14-s − 0.00117·16-s − 0.670·17-s + 0.418·19-s − 2.76·20-s + 1.62·23-s + 1.91·25-s − 0.891·26-s + 2.61·28-s + 0.123·29-s − 1.13·31-s + 1.00·32-s + 1.08·34-s − 2.76·35-s − 1.68·37-s − 0.676·38-s + 1.70·40-s + 0.171·41-s − 0.133·43-s − 2.62·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 9.15T + 32T^{2} \) |
| 5 | \( 1 + 95.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 209.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 335.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 799.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 658.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 559.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.58e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.45e5T + 8.58e9T^{2} \) |
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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
−8.556944013275636942512580245361, −8.141729495089878717236852976326, −7.35042282636431104999567642824, −6.89625403015730318926031593969, −5.16846547441704388568521937345, −4.36520472274302912614185879857, −3.23191812437331315284874302780, −1.79829985467128187227936462791, −0.978290303741302718985989703204, 0,
0.978290303741302718985989703204, 1.79829985467128187227936462791, 3.23191812437331315284874302780, 4.36520472274302912614185879857, 5.16846547441704388568521937345, 6.89625403015730318926031593969, 7.35042282636431104999567642824, 8.141729495089878717236852976326, 8.556944013275636942512580245361
