| L(s) = 1 | − 579.·2-s + 2.05e5·4-s − 3.27e3·5-s − 5.76e6·7-s − 4.29e7·8-s + 1.89e6·10-s + 5.18e8·11-s − 2.75e9·13-s + 3.34e9·14-s − 1.99e9·16-s − 1.79e10·17-s − 9.37e9·19-s − 6.71e8·20-s − 3.00e11·22-s + 2.27e11·23-s − 7.62e11·25-s + 1.59e12·26-s − 1.18e12·28-s + 4.34e12·29-s − 3.53e10·31-s + 6.78e12·32-s + 1.03e13·34-s + 1.88e10·35-s + 1.79e13·37-s + 5.43e12·38-s + 1.40e11·40-s − 5.22e12·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.56·4-s − 0.00375·5-s − 0.377·7-s − 0.904·8-s + 0.00600·10-s + 0.728·11-s − 0.935·13-s + 0.605·14-s − 0.116·16-s − 0.623·17-s − 0.126·19-s − 0.00586·20-s − 1.16·22-s + 0.605·23-s − 0.999·25-s + 1.49·26-s − 0.591·28-s + 1.61·29-s − 0.00744·31-s + 1.09·32-s + 0.998·34-s + 0.00141·35-s + 0.838·37-s + 0.202·38-s + 0.00339·40-s − 0.102·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.6146303605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6146303605\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 5.76e6T \) |
| good | 2 | \( 1 + 579.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 3.27e3T + 7.62e11T^{2} \) |
| 11 | \( 1 - 5.18e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.75e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.79e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 9.37e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.27e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 4.34e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.53e10T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.79e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.22e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 6.97e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.91e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.13e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.09e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.01e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.44e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.36e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.63e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.89e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.88e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 6.24e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.04e14T + 5.95e33T^{2} \) |
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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
−11.23202034378983697937839632222, −10.08966273152876605542953512869, −9.358231937257763176780567268600, −8.370991395246239361038924524174, −7.23995308930125394743415691185, −6.33446125424073408785210671815, −4.51717135645936269200949481792, −2.80008190383350025272371690861, −1.62976185047579186970647069342, −0.47488641455269438027130746846,
0.47488641455269438027130746846, 1.62976185047579186970647069342, 2.80008190383350025272371690861, 4.51717135645936269200949481792, 6.33446125424073408785210671815, 7.23995308930125394743415691185, 8.370991395246239361038924524174, 9.358231937257763176780567268600, 10.08966273152876605542953512869, 11.23202034378983697937839632222
