| L(s) = 1 | + 0.246·2-s − 1.93·4-s − 0.972·8-s − 3·9-s − 6.04·11-s + 3.63·16-s − 0.740·18-s − 1.49·22-s + 0.850·23-s − 5·25-s − 7.02·29-s + 2.84·32-s + 5.81·36-s − 8.98·37-s + 13.1·43-s + 11.7·44-s + 0.210·46-s − 1.23·50-s + 13.6·53-s − 1.73·58-s − 6.57·64-s + 14.9·67-s − 12.1·71-s + 2.91·72-s − 2.21·74-s − 14.0·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.174·2-s − 0.969·4-s − 0.343·8-s − 9-s − 1.82·11-s + 0.909·16-s − 0.174·18-s − 0.318·22-s + 0.177·23-s − 25-s − 1.30·29-s + 0.502·32-s + 0.969·36-s − 1.47·37-s + 1.99·43-s + 1.76·44-s + 0.0309·46-s − 0.174·50-s + 1.86·53-s − 0.227·58-s − 0.821·64-s + 1.82·67-s − 1.43·71-s + 0.343·72-s − 0.257·74-s − 1.58·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.850T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13.1T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.98182200365516258508268747942, −10.15328161263522334241712262199, −9.119959185987997874458119466685, −8.293490577303394711747399990243, −7.46583979773450980467251956817, −5.70402459824507246282884564819, −5.28405997773415472992370215891, −3.88499265972104844070750824198, −2.60735752415256012047946377410, 0,
2.60735752415256012047946377410, 3.88499265972104844070750824198, 5.28405997773415472992370215891, 5.70402459824507246282884564819, 7.46583979773450980467251956817, 8.293490577303394711747399990243, 9.119959185987997874458119466685, 10.15328161263522334241712262199, 10.98182200365516258508268747942
