L(s) = 1 | − 2.68e4·2-s + 3.95e8·3-s − 3.36e10·4-s + 8.21e11·5-s − 1.06e13·6-s + 6.06e14·7-s + 1.82e15·8-s + 1.06e17·9-s − 2.20e16·10-s + 1.23e18·11-s − 1.33e19·12-s − 8.46e17·13-s − 1.62e19·14-s + 3.25e20·15-s + 1.10e21·16-s − 3.92e21·17-s − 2.85e21·18-s − 3.52e20·19-s − 2.76e22·20-s + 2.40e23·21-s − 3.30e22·22-s − 8.58e23·23-s + 7.21e23·24-s − 2.23e24·25-s + 2.26e22·26-s + 2.23e25·27-s − 2.04e25·28-s + ⋯ |
L(s) = 1 | − 0.144·2-s + 1.76·3-s − 0.979·4-s + 0.481·5-s − 0.255·6-s + 0.986·7-s + 0.286·8-s + 2.13·9-s − 0.0696·10-s + 0.735·11-s − 1.73·12-s − 0.0271·13-s − 0.142·14-s + 0.851·15-s + 0.937·16-s − 1.14·17-s − 0.308·18-s − 0.0147·19-s − 0.471·20-s + 1.74·21-s − 0.106·22-s − 1.26·23-s + 0.506·24-s − 0.768·25-s + 0.00392·26-s + 1.99·27-s − 0.965·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+35/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(18)\) |
\(\approx\) |
\(2.558058893\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.558058893\) |
\(L(\frac{37}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 + 2.68e4T + 3.43e10T^{2} \) |
| 3 | \( 1 - 3.95e8T + 5.00e16T^{2} \) |
| 5 | \( 1 - 8.21e11T + 2.91e24T^{2} \) |
| 7 | \( 1 - 6.06e14T + 3.78e29T^{2} \) |
| 11 | \( 1 - 1.23e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 8.46e17T + 9.72e38T^{2} \) |
| 17 | \( 1 + 3.92e21T + 1.16e43T^{2} \) |
| 19 | \( 1 + 3.52e20T + 5.70e44T^{2} \) |
| 23 | \( 1 + 8.58e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 1.38e25T + 1.52e51T^{2} \) |
| 31 | \( 1 + 3.33e25T + 1.57e52T^{2} \) |
| 37 | \( 1 - 1.99e27T + 7.71e54T^{2} \) |
| 41 | \( 1 - 2.10e28T + 2.80e56T^{2} \) |
| 43 | \( 1 + 4.57e28T + 1.48e57T^{2} \) |
| 47 | \( 1 + 1.72e29T + 3.33e58T^{2} \) |
| 53 | \( 1 - 1.81e30T + 2.23e60T^{2} \) |
| 59 | \( 1 + 5.02e30T + 9.54e61T^{2} \) |
| 61 | \( 1 + 3.25e29T + 3.06e62T^{2} \) |
| 67 | \( 1 + 4.01e31T + 8.17e63T^{2} \) |
| 71 | \( 1 - 3.85e32T + 6.22e64T^{2} \) |
| 73 | \( 1 - 9.93e31T + 1.64e65T^{2} \) |
| 79 | \( 1 + 2.62e33T + 2.61e66T^{2} \) |
| 83 | \( 1 - 6.17e33T + 1.47e67T^{2} \) |
| 89 | \( 1 - 6.75e33T + 1.69e68T^{2} \) |
| 97 | \( 1 + 8.10e34T + 3.44e69T^{2} \) |
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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
−24.51213991249091562445044194612, −21.60990521007096499237175467177, −19.84115990496816626692568244831, −18.00626150863511425985242461230, −14.63850500192579912510421319527, −13.54187465949206101269929955780, −9.459703826379898497888748758919, −8.157521796120929573693857359706, −4.13950803345000955418253124047, −1.83589775632061795320516641947,
1.83589775632061795320516641947, 4.13950803345000955418253124047, 8.157521796120929573693857359706, 9.459703826379898497888748758919, 13.54187465949206101269929955780, 14.63850500192579912510421319527, 18.00626150863511425985242461230, 19.84115990496816626692568244831, 21.60990521007096499237175467177, 24.51213991249091562445044194612