| L(s) = 1 | − 5.31·2-s − 3.75·4-s + 23.8·5-s − 11.7·7-s + 190.·8-s − 126.·10-s − 280.·11-s − 835.·13-s + 62.6·14-s − 889.·16-s + 1.80e3·17-s + 2.35e3·19-s − 89.6·20-s + 1.49e3·22-s + 529·23-s − 2.55e3·25-s + 4.43e3·26-s + 44.2·28-s + 1.20e3·29-s − 3.39e3·31-s − 1.35e3·32-s − 9.57e3·34-s − 281.·35-s + 8.35e3·37-s − 1.25e4·38-s + 4.53e3·40-s + 1.07e4·41-s + ⋯ |
| L(s) = 1 | − 0.939·2-s − 0.117·4-s + 0.427·5-s − 0.0908·7-s + 1.04·8-s − 0.401·10-s − 0.699·11-s − 1.37·13-s + 0.0854·14-s − 0.868·16-s + 1.51·17-s + 1.49·19-s − 0.0501·20-s + 0.657·22-s + 0.208·23-s − 0.817·25-s + 1.28·26-s + 0.0106·28-s + 0.266·29-s − 0.634·31-s − 0.233·32-s − 1.42·34-s − 0.0388·35-s + 1.00·37-s − 1.40·38-s + 0.448·40-s + 0.994·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 5.31T + 32T^{2} \) |
| 5 | \( 1 - 23.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 11.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 280.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 835.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.35e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 803.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.87e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.41e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67063856894861394536832977627, −9.708871028486939111938469184529, −9.441784536622314042732193560585, −7.82192275392581278465586341785, −7.50501459089048137332854230968, −5.71543780375157358835903748813, −4.73682209028108269553044879817, −2.96039102310691861634363863252, −1.39300401348241152523219268271, 0,
1.39300401348241152523219268271, 2.96039102310691861634363863252, 4.73682209028108269553044879817, 5.71543780375157358835903748813, 7.50501459089048137332854230968, 7.82192275392581278465586341785, 9.441784536622314042732193560585, 9.708871028486939111938469184529, 10.67063856894861394536832977627
