| L(s) = 1 | + 2.64·2-s − 2.38·3-s + 4.98·4-s + 4.31·5-s − 6.30·6-s + 3.16·7-s + 7.89·8-s + 2.68·9-s + 11.4·10-s − 1.50·11-s − 11.8·12-s + 13-s + 8.36·14-s − 10.2·15-s + 10.8·16-s − 1.79·17-s + 7.10·18-s − 4.70·19-s + 21.5·20-s − 7.54·21-s − 3.98·22-s − 7.11·23-s − 18.8·24-s + 13.6·25-s + 2.64·26-s + 0.741·27-s + 15.7·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 1.37·3-s + 2.49·4-s + 1.92·5-s − 2.57·6-s + 1.19·7-s + 2.79·8-s + 0.896·9-s + 3.60·10-s − 0.454·11-s − 3.43·12-s + 0.277·13-s + 2.23·14-s − 2.65·15-s + 2.72·16-s − 0.434·17-s + 1.67·18-s − 1.08·19-s + 4.80·20-s − 1.64·21-s − 0.848·22-s − 1.48·23-s − 3.84·24-s + 2.72·25-s + 0.518·26-s + 0.142·27-s + 2.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.992080437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.992080437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 - 1.72T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 - 7.96T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 - 0.764T + 61T^{2} \) |
| 67 | \( 1 - 0.946T + 67T^{2} \) |
| 71 | \( 1 + 5.08T + 71T^{2} \) |
| 73 | \( 1 + 8.25T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 - 3.19T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−10.28158728942188578426805941748, −8.851865439557929450962523953338, −7.45922835134882610290174199707, −6.48839798275813556522082512199, −5.94781279403034908992519688342, −5.39243324560424675782622044127, −4.95784006506676678672136864580, −3.92977918367374976741866215595, −2.19891680046777893210690031723, −1.78826389114702445141366322353,
1.78826389114702445141366322353, 2.19891680046777893210690031723, 3.92977918367374976741866215595, 4.95784006506676678672136864580, 5.39243324560424675782622044127, 5.94781279403034908992519688342, 6.48839798275813556522082512199, 7.45922835134882610290174199707, 8.851865439557929450962523953338, 10.28158728942188578426805941748
