| L(s) = 1 | + 1.29e3·2-s − 4.24e5·4-s − 9.76e6·5-s + 1.27e9·7-s − 3.26e9·8-s − 1.26e10·10-s + 1.21e11·11-s + 7.37e11·13-s + 1.65e12·14-s − 3.32e12·16-s − 6.05e12·17-s − 3.23e13·19-s + 4.14e12·20-s + 1.56e14·22-s + 3.98e13·23-s + 9.53e13·25-s + 9.53e14·26-s − 5.43e14·28-s − 1.40e15·29-s − 2.62e15·31-s + 2.53e15·32-s − 7.83e15·34-s − 1.24e16·35-s − 3.44e15·37-s − 4.18e16·38-s + 3.18e16·40-s − 7.99e16·41-s + ⋯ |
| L(s) = 1 | + 0.893·2-s − 0.202·4-s − 0.447·5-s + 1.71·7-s − 1.07·8-s − 0.399·10-s + 1.40·11-s + 1.48·13-s + 1.52·14-s − 0.756·16-s − 0.728·17-s − 1.20·19-s + 0.0904·20-s + 1.25·22-s + 0.200·23-s + 0.199·25-s + 1.32·26-s − 0.346·28-s − 0.622·29-s − 0.575·31-s + 0.398·32-s − 0.650·34-s − 0.765·35-s − 0.117·37-s − 1.08·38-s + 0.480·40-s − 0.929·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.891412489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.891412489\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 2 | \( 1 - 1.29e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.27e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.21e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.37e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.05e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.23e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.98e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.40e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.62e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.44e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.99e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.07e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.95e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.33e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.93e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.99e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 9.21e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.75e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.81e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.11e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 3.39e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.38e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.12e20T + 5.27e41T^{2} \) |
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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
−11.57769116536189332211104335016, −11.02639982881412156552774520729, −8.939949764828638717794415421509, −8.366474792681122651045395315403, −6.69661646011760205096314348576, −5.48762596346637760272339649288, −4.25595363719694055877070508889, −3.83725941632738691745049331827, −1.98420057947417468079924546350, −0.838567829295619169433666960052,
0.838567829295619169433666960052, 1.98420057947417468079924546350, 3.83725941632738691745049331827, 4.25595363719694055877070508889, 5.48762596346637760272339649288, 6.69661646011760205096314348576, 8.366474792681122651045395315403, 8.939949764828638717794415421509, 11.02639982881412156552774520729, 11.57769116536189332211104335016
