| L(s) = 1 | − 1.23·3-s − 3.39·5-s + 7-s − 1.48·9-s − 11-s − 13-s + 4.18·15-s + 3.16·17-s − 2.19·19-s − 1.23·21-s + 6.34·23-s + 6.53·25-s + 5.52·27-s + 1.18·29-s − 0.106·31-s + 1.23·33-s − 3.39·35-s + 8.25·37-s + 1.23·39-s + 2.65·41-s − 10.2·43-s + 5.04·45-s − 1.67·47-s + 49-s − 3.89·51-s − 1.94·53-s + 3.39·55-s + ⋯ |
| L(s) = 1 | − 0.710·3-s − 1.51·5-s + 0.377·7-s − 0.495·9-s − 0.301·11-s − 0.277·13-s + 1.07·15-s + 0.767·17-s − 0.503·19-s − 0.268·21-s + 1.32·23-s + 1.30·25-s + 1.06·27-s + 0.219·29-s − 0.0191·31-s + 0.214·33-s − 0.574·35-s + 1.35·37-s + 0.197·39-s + 0.414·41-s − 1.56·43-s + 0.751·45-s − 0.244·47-s + 0.142·49-s − 0.545·51-s − 0.266·53-s + 0.458·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + 0.106T + 31T^{2} \) |
| 37 | \( 1 - 8.25T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 8.62T + 97T^{2} \) |
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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
−8.143654152946322878852656213765, −7.36534887070987558016925433171, −6.72430689115741806327896966777, −5.76280584338948468459544870429, −4.99964252123126366875296625046, −4.42066714566734548532905157294, −3.44584926093316005366426671634, −2.67260380322195818599227912329, −1.04773499514804901745420932284, 0,
1.04773499514804901745420932284, 2.67260380322195818599227912329, 3.44584926093316005366426671634, 4.42066714566734548532905157294, 4.99964252123126366875296625046, 5.76280584338948468459544870429, 6.72430689115741806327896966777, 7.36534887070987558016925433171, 8.143654152946322878852656213765
