| L(s) = 1 | − 2.18·2-s + 1.75·3-s + 2.76·4-s − 2.11·5-s − 3.83·6-s − 1.67·8-s + 0.0920·9-s + 4.60·10-s + 5.76·11-s + 4.86·12-s − 3.71·15-s − 1.87·16-s + 1.64·17-s − 0.200·18-s + 2.67·19-s − 5.83·20-s − 12.5·22-s + 6.42·23-s − 2.94·24-s − 0.545·25-s − 5.11·27-s − 6.04·29-s + 8.10·30-s − 5.12·31-s + 7.45·32-s + 10.1·33-s − 3.58·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.01·3-s + 1.38·4-s − 0.943·5-s − 1.56·6-s − 0.591·8-s + 0.0306·9-s + 1.45·10-s + 1.73·11-s + 1.40·12-s − 0.958·15-s − 0.469·16-s + 0.397·17-s − 0.0473·18-s + 0.612·19-s − 1.30·20-s − 2.68·22-s + 1.33·23-s − 0.600·24-s − 0.109·25-s − 0.984·27-s − 1.12·29-s + 1.47·30-s − 0.919·31-s + 1.31·32-s + 1.76·33-s − 0.614·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−7.65985569915888670099895127112, −7.18651021196107866971422755734, −6.56133468570494397757718772183, −5.49339860643013819444480724305, −4.36101440480386101698805551694, −3.59438862622294930673654970945, −3.08885952624629299592861918305, −1.87706057943070898978876178820, −1.22397620654511958981845069706, 0,
1.22397620654511958981845069706, 1.87706057943070898978876178820, 3.08885952624629299592861918305, 3.59438862622294930673654970945, 4.36101440480386101698805551694, 5.49339860643013819444480724305, 6.56133468570494397757718772183, 7.18651021196107866971422755734, 7.65985569915888670099895127112
