| L(s) = 1 | + 2.09e4·2-s + 1.43e7·3-s − 1.70e9·4-s + 4.40e10·5-s + 3.00e11·6-s − 1.09e13·7-s − 8.07e13·8-s + 2.05e14·9-s + 9.22e14·10-s − 1.20e16·11-s − 2.45e16·12-s + 6.41e16·13-s − 2.29e17·14-s + 6.32e17·15-s + 1.98e18·16-s − 1.57e19·17-s + 4.30e18·18-s − 1.01e20·19-s − 7.53e19·20-s − 1.57e20·21-s − 2.52e20·22-s − 1.16e21·23-s − 1.15e21·24-s − 2.71e21·25-s + 1.34e21·26-s + 2.95e21·27-s + 1.87e22·28-s + ⋯ |
| L(s) = 1 | + 0.451·2-s + 0.577·3-s − 0.795·4-s + 0.645·5-s + 0.260·6-s − 0.873·7-s − 0.811·8-s + 0.333·9-s + 0.291·10-s − 0.872·11-s − 0.459·12-s + 0.347·13-s − 0.394·14-s + 0.372·15-s + 0.429·16-s − 1.33·17-s + 0.150·18-s − 1.53·19-s − 0.513·20-s − 0.504·21-s − 0.393·22-s − 0.911·23-s − 0.468·24-s − 0.583·25-s + 0.157·26-s + 0.192·27-s + 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.43e7T \) |
| good | 2 | \( 1 - 2.09e4T + 2.14e9T^{2} \) |
| 5 | \( 1 - 4.40e10T + 4.65e21T^{2} \) |
| 7 | \( 1 + 1.09e13T + 1.57e26T^{2} \) |
| 11 | \( 1 + 1.20e16T + 1.91e32T^{2} \) |
| 13 | \( 1 - 6.41e16T + 3.40e34T^{2} \) |
| 17 | \( 1 + 1.57e19T + 1.39e38T^{2} \) |
| 19 | \( 1 + 1.01e20T + 4.37e39T^{2} \) |
| 23 | \( 1 + 1.16e21T + 1.63e42T^{2} \) |
| 29 | \( 1 + 1.53e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 2.39e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 5.11e22T + 4.11e48T^{2} \) |
| 41 | \( 1 + 1.02e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 9.54e24T + 4.34e50T^{2} \) |
| 47 | \( 1 - 1.30e26T + 6.83e51T^{2} \) |
| 53 | \( 1 + 3.91e26T + 2.83e53T^{2} \) |
| 59 | \( 1 - 5.34e27T + 7.87e54T^{2} \) |
| 61 | \( 1 - 1.24e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 3.44e28T + 4.05e56T^{2} \) |
| 71 | \( 1 - 3.70e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 1.06e29T + 5.79e57T^{2} \) |
| 79 | \( 1 - 2.48e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 8.38e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 3.06e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 2.89e30T + 3.88e61T^{2} \) |
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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
−17.68551850013125804367001956647, −15.49657468927713739582340493583, −13.73371302310480820357687378119, −12.90048748399637768454386053105, −10.03845988698785089401898209793, −8.605956664172178736323373405562, −6.13042128831849306310337388082, −4.20995343350361975982128564642, −2.48482734638277725353405184833, 0,
2.48482734638277725353405184833, 4.20995343350361975982128564642, 6.13042128831849306310337388082, 8.605956664172178736323373405562, 10.03845988698785089401898209793, 12.90048748399637768454386053105, 13.73371302310480820357687378119, 15.49657468927713739582340493583, 17.68551850013125804367001956647
