| L(s) = 1 | + 10.2·2-s + 73.1·4-s − 26.5·5-s − 49·7-s + 421.·8-s − 272.·10-s − 465.·11-s + 169·13-s − 502.·14-s + 1.98e3·16-s + 901.·17-s − 1.76e3·19-s − 1.94e3·20-s − 4.77e3·22-s + 4.43e3·23-s − 2.41e3·25-s + 1.73e3·26-s − 3.58e3·28-s − 7.08e3·29-s − 6.01e3·31-s + 6.82e3·32-s + 9.24e3·34-s + 1.30e3·35-s + 7.66e3·37-s − 1.80e4·38-s − 1.12e4·40-s − 8.35e3·41-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.28·4-s − 0.475·5-s − 0.377·7-s + 2.32·8-s − 0.862·10-s − 1.16·11-s + 0.277·13-s − 0.685·14-s + 1.93·16-s + 0.756·17-s − 1.11·19-s − 1.08·20-s − 2.10·22-s + 1.74·23-s − 0.773·25-s + 0.502·26-s − 0.863·28-s − 1.56·29-s − 1.12·31-s + 1.17·32-s + 1.37·34-s + 0.179·35-s + 0.920·37-s − 2.02·38-s − 1.10·40-s − 0.776·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 49T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 5 | \( 1 + 26.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 465.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 901.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.76e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.47e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.60e4T + 8.58e9T^{2} \) |
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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
−9.023940385352428100993579697525, −7.75397833138899212290435598307, −7.17890828676750546547992636594, −6.09912679219111989553448297753, −5.41868082594935000812116179021, −4.56116920214383592018531748205, −3.59085159221012288935590809155, −2.92994622032871166566664776465, −1.78775740448568128733343918753, 0,
1.78775740448568128733343918753, 2.92994622032871166566664776465, 3.59085159221012288935590809155, 4.56116920214383592018531748205, 5.41868082594935000812116179021, 6.09912679219111989553448297753, 7.17890828676750546547992636594, 7.75397833138899212290435598307, 9.023940385352428100993579697525
