| L(s) = 1 | − 0.528·3-s + 0.933·5-s + 7-s − 2.72·9-s − 11-s − 13-s − 0.493·15-s + 0.372·17-s − 3.48·19-s − 0.528·21-s − 4.94·23-s − 4.12·25-s + 3.02·27-s + 9.77·29-s + 8.42·31-s + 0.528·33-s + 0.933·35-s − 2.32·37-s + 0.528·39-s + 8.25·41-s + 1.29·43-s − 2.54·45-s + 10.0·47-s + 49-s − 0.196·51-s + 1.68·53-s − 0.933·55-s + ⋯ |
| L(s) = 1 | − 0.305·3-s + 0.417·5-s + 0.377·7-s − 0.906·9-s − 0.301·11-s − 0.277·13-s − 0.127·15-s + 0.0902·17-s − 0.798·19-s − 0.115·21-s − 1.03·23-s − 0.825·25-s + 0.582·27-s + 1.81·29-s + 1.51·31-s + 0.0920·33-s + 0.157·35-s − 0.381·37-s + 0.0846·39-s + 1.28·41-s + 0.197·43-s − 0.378·45-s + 1.46·47-s + 0.142·49-s − 0.0275·51-s + 0.230·53-s − 0.125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.528T + 3T^{2} \) |
| 5 | \( 1 - 0.933T + 5T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 - 8.25T + 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 9.57T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 + 2.45T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−7.53275365164324855665687720537, −6.68660214580312491874438502051, −5.89692603207504055247203582393, −5.66657410174784872018501670798, −4.60045910738049366441115992440, −4.14357099756275777018877441698, −2.79718030699381078582106173845, −2.42343407979205093103505381644, −1.21392015727861228154990853987, 0,
1.21392015727861228154990853987, 2.42343407979205093103505381644, 2.79718030699381078582106173845, 4.14357099756275777018877441698, 4.60045910738049366441115992440, 5.66657410174784872018501670798, 5.89692603207504055247203582393, 6.68660214580312491874438502051, 7.53275365164324855665687720537
