| L(s) = 1 | + 9.16·2-s + 52·4-s − 87.0·5-s + 49·7-s + 183.·8-s − 798·10-s + 32.0·11-s − 940·13-s + 449.·14-s + 15.9·16-s − 9.16·17-s − 1.93e3·19-s − 4.52e3·20-s + 294·22-s + 2.05e3·23-s + 4.45e3·25-s − 8.61e3·26-s + 2.54e3·28-s − 5.82e3·29-s − 8.69e3·31-s − 5.71e3·32-s − 84·34-s − 4.26e3·35-s − 2.45e3·37-s − 1.77e4·38-s − 1.59e4·40-s + 1.16e4·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.62·4-s − 1.55·5-s + 0.377·7-s + 1.01·8-s − 2.52·10-s + 0.0799·11-s − 1.54·13-s + 0.612·14-s + 0.0156·16-s − 0.00769·17-s − 1.23·19-s − 2.53·20-s + 0.129·22-s + 0.811·23-s + 1.42·25-s − 2.49·26-s + 0.614·28-s − 1.28·29-s − 1.62·31-s − 0.987·32-s − 0.0124·34-s − 0.588·35-s − 0.294·37-s − 1.99·38-s − 1.57·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 - 9.16T + 32T^{2} \) |
| 5 | \( 1 + 87.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 32.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 940T + 3.71e5T^{2} \) |
| 17 | \( 1 + 9.16T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.16e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.79e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
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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
−11.44390017883523439631601282005, −10.86975771607368338251712114889, −9.055367154766008429727857964414, −7.64331450978487001330459764200, −7.00599845431248170224029506877, −5.46030857894121274983000330035, −4.46825846646290614771008104312, −3.73473984255377656919125477981, −2.38795921660290718662933992932, 0,
2.38795921660290718662933992932, 3.73473984255377656919125477981, 4.46825846646290614771008104312, 5.46030857894121274983000330035, 7.00599845431248170224029506877, 7.64331450978487001330459764200, 9.055367154766008429727857964414, 10.86975771607368338251712114889, 11.44390017883523439631601282005
