| L(s) = 1 | − 1.25·2-s + 1.22·3-s − 0.425·4-s + 5-s − 1.54·6-s − 7-s + 3.04·8-s − 1.48·9-s − 1.25·10-s + 2.28·11-s − 0.523·12-s + 0.818·13-s + 1.25·14-s + 1.22·15-s − 2.96·16-s − 3.37·17-s + 1.86·18-s − 2.87·19-s − 0.425·20-s − 1.22·21-s − 2.87·22-s + 4.95·23-s + 3.74·24-s + 25-s − 1.02·26-s − 5.51·27-s + 0.425·28-s + ⋯ |
| L(s) = 1 | − 0.887·2-s + 0.709·3-s − 0.212·4-s + 0.447·5-s − 0.629·6-s − 0.377·7-s + 1.07·8-s − 0.496·9-s − 0.396·10-s + 0.689·11-s − 0.151·12-s + 0.226·13-s + 0.335·14-s + 0.317·15-s − 0.741·16-s − 0.817·17-s + 0.440·18-s − 0.658·19-s − 0.0951·20-s − 0.268·21-s − 0.611·22-s + 1.03·23-s + 0.763·24-s + 0.200·25-s − 0.201·26-s − 1.06·27-s + 0.0804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 3 | \( 1 - 1.22T + 3T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 - 0.818T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 + 6.97T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 8.22T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 - 2.91T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.63982008419642382225738337566, −7.00629013713652664755112060495, −6.21469540998230528333612624833, −5.51600115977465015921850851101, −4.48119542748135115978808315683, −3.90561443861395798814167777203, −2.92040543126396822652047148446, −2.13596604113049885237817541165, −1.20910059933068961156396122328, 0,
1.20910059933068961156396122328, 2.13596604113049885237817541165, 2.92040543126396822652047148446, 3.90561443861395798814167777203, 4.48119542748135115978808315683, 5.51600115977465015921850851101, 6.21469540998230528333612624833, 7.00629013713652664755112060495, 7.63982008419642382225738337566
