| L(s) = 1 | − 3.21·2-s − 3·3-s + 2.35·4-s + 9.65·6-s + 7·7-s + 18.1·8-s + 9·9-s − 2.09·11-s − 7.05·12-s − 80.8·13-s − 22.5·14-s − 77.2·16-s − 101.·17-s − 28.9·18-s + 143.·19-s − 21·21-s + 6.75·22-s + 116.·23-s − 54.5·24-s + 260.·26-s − 27·27-s + 16.4·28-s + 181.·29-s + 303.·31-s + 103.·32-s + 6.29·33-s + 328.·34-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.293·4-s + 0.656·6-s + 0.377·7-s + 0.803·8-s + 0.333·9-s − 0.0575·11-s − 0.169·12-s − 1.72·13-s − 0.429·14-s − 1.20·16-s − 1.45·17-s − 0.379·18-s + 1.73·19-s − 0.218·21-s + 0.0654·22-s + 1.05·23-s − 0.463·24-s + 1.96·26-s − 0.192·27-s + 0.111·28-s + 1.16·29-s + 1.75·31-s + 0.570·32-s + 0.0332·33-s + 1.65·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 3.21T + 8T^{2} \) |
| 11 | \( 1 + 2.09T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 379.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 43.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 812.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 471.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 886.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 134.T + 9.12e5T^{2} \) |
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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
−9.944504843970747104902750692049, −9.243551529370470552676376470385, −8.290947725686431985612209910351, −7.35377986734571396282747206603, −6.72342074589394933126000025308, −5.07595406855566013906505098822, −4.61645302411216913350267400647, −2.65294018966755217437324764249, −1.21046594203842898808588622703, 0,
1.21046594203842898808588622703, 2.65294018966755217437324764249, 4.61645302411216913350267400647, 5.07595406855566013906505098822, 6.72342074589394933126000025308, 7.35377986734571396282747206603, 8.290947725686431985612209910351, 9.243551529370470552676376470385, 9.944504843970747104902750692049
