| L(s) = 1 | + 2-s − 1.99·3-s + 4-s + 3.02·5-s − 1.99·6-s − 1.37·7-s + 8-s + 0.964·9-s + 3.02·10-s + 0.397·11-s − 1.99·12-s − 4.55·13-s − 1.37·14-s − 6.02·15-s + 16-s − 3.23·17-s + 0.964·18-s − 19-s + 3.02·20-s + 2.73·21-s + 0.397·22-s + 3.58·23-s − 1.99·24-s + 4.16·25-s − 4.55·26-s + 4.05·27-s − 1.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s + 1.35·5-s − 0.812·6-s − 0.519·7-s + 0.353·8-s + 0.321·9-s + 0.957·10-s + 0.119·11-s − 0.574·12-s − 1.26·13-s − 0.367·14-s − 1.55·15-s + 0.250·16-s − 0.785·17-s + 0.227·18-s − 0.229·19-s + 0.677·20-s + 0.596·21-s + 0.0846·22-s + 0.746·23-s − 0.406·24-s + 0.833·25-s − 0.893·26-s + 0.780·27-s − 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 - 0.397T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 9.38T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 - 0.515T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 - 0.464T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 + 6.52T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
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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
−6.97840704356413137524972434975, −6.60386281647385987386965441005, −6.04939749821459101934216775463, −5.37211065883619017943903947469, −4.94794548514841117749847049815, −4.19051142217301549285635031139, −2.90842474808664608613756533819, −2.39624117093668334238106532155, −1.33276998885765438841344402177, 0,
1.33276998885765438841344402177, 2.39624117093668334238106532155, 2.90842474808664608613756533819, 4.19051142217301549285635031139, 4.94794548514841117749847049815, 5.37211065883619017943903947469, 6.04939749821459101934216775463, 6.60386281647385987386965441005, 6.97840704356413137524972434975
