| L(s) = 1 | + 3-s − 1.29·5-s + 2.07·7-s + 9-s − 3.34·11-s + 2.39·13-s − 1.29·15-s + 4.56·17-s − 2.47·19-s + 2.07·21-s + 23-s − 3.32·25-s + 27-s + 29-s − 10.5·31-s − 3.34·33-s − 2.68·35-s − 0.207·37-s + 2.39·39-s − 8.96·41-s − 7.63·43-s − 1.29·45-s − 0.402·47-s − 2.70·49-s + 4.56·51-s − 11.7·53-s + 4.33·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.579·5-s + 0.783·7-s + 0.333·9-s − 1.00·11-s + 0.664·13-s − 0.334·15-s + 1.10·17-s − 0.568·19-s + 0.452·21-s + 0.208·23-s − 0.664·25-s + 0.192·27-s + 0.185·29-s − 1.89·31-s − 0.582·33-s − 0.453·35-s − 0.0341·37-s + 0.383·39-s − 1.40·41-s − 1.16·43-s − 0.193·45-s − 0.0586·47-s − 0.386·49-s + 0.639·51-s − 1.60·53-s + 0.584·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 - 2.07T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.207T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 + 7.63T + 43T^{2} \) |
| 47 | \( 1 + 0.402T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 - 8.77T + 71T^{2} \) |
| 73 | \( 1 - 2.09T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−7.68835769474813816882526818220, −7.02430591709496586682066135471, −6.07671619225973810017825289848, −5.23447341534752719654548010819, −4.73698724256918459729658165974, −3.65283509783950124174739592151, −3.33751666564754310903987238677, −2.15919536834616792503931305142, −1.43558036318543306773175639961, 0,
1.43558036318543306773175639961, 2.15919536834616792503931305142, 3.33751666564754310903987238677, 3.65283509783950124174739592151, 4.73698724256918459729658165974, 5.23447341534752719654548010819, 6.07671619225973810017825289848, 7.02430591709496586682066135471, 7.68835769474813816882526818220
