| L(s) = 1 | + 2.56·2-s + 1.12·3-s + 4.57·4-s + 1.24·5-s + 2.88·6-s + 2.50·7-s + 6.60·8-s − 1.73·9-s + 3.20·10-s + 11-s + 5.15·12-s + 6.42·14-s + 1.40·15-s + 7.78·16-s + 2.78·17-s − 4.43·18-s − 5.79·19-s + 5.70·20-s + 2.82·21-s + 2.56·22-s − 8.60·23-s + 7.44·24-s − 3.44·25-s − 5.33·27-s + 11.4·28-s − 0.584·29-s + 3.60·30-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.650·3-s + 2.28·4-s + 0.558·5-s + 1.17·6-s + 0.946·7-s + 2.33·8-s − 0.576·9-s + 1.01·10-s + 0.301·11-s + 1.48·12-s + 1.71·14-s + 0.363·15-s + 1.94·16-s + 0.676·17-s − 1.04·18-s − 1.32·19-s + 1.27·20-s + 0.615·21-s + 0.546·22-s − 1.79·23-s + 1.51·24-s − 0.688·25-s − 1.02·27-s + 2.16·28-s − 0.108·29-s + 0.658·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.291850241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.291850241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 + 8.60T + 23T^{2} \) |
| 29 | \( 1 + 0.584T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 - 0.167T + 41T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.39T + 53T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.69T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 0.577T + 89T^{2} \) |
| 97 | \( 1 + 3.91T + 97T^{2} \) |
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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
−9.228683244034658195551924562858, −8.116929899811195018990266400316, −7.70153515466961312643525646465, −6.40239919969037187005062020485, −5.90324039932996144543463661479, −5.15819517075313412821505476612, −4.17599570516765756930959501161, −3.58770445103657965574502239626, −2.34241904051139675505698096420, −1.91048050542531803716997485783,
1.91048050542531803716997485783, 2.34241904051139675505698096420, 3.58770445103657965574502239626, 4.17599570516765756930959501161, 5.15819517075313412821505476612, 5.90324039932996144543463661479, 6.40239919969037187005062020485, 7.70153515466961312643525646465, 8.116929899811195018990266400316, 9.228683244034658195551924562858
