| L(s) = 1 | + 2.23·2-s − 0.546·3-s + 3.00·4-s − 3.24·5-s − 1.22·6-s + 2.54·7-s + 2.25·8-s − 2.70·9-s − 7.25·10-s − 4.12·11-s − 1.64·12-s + 13-s + 5.70·14-s + 1.77·15-s − 0.973·16-s − 4.63·17-s − 6.04·18-s − 3.00·19-s − 9.74·20-s − 1.39·21-s − 9.23·22-s − 1.11·23-s − 1.23·24-s + 5.50·25-s + 2.23·26-s + 3.11·27-s + 7.66·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.315·3-s + 1.50·4-s − 1.44·5-s − 0.499·6-s + 0.963·7-s + 0.796·8-s − 0.900·9-s − 2.29·10-s − 1.24·11-s − 0.474·12-s + 0.277·13-s + 1.52·14-s + 0.457·15-s − 0.243·16-s − 1.12·17-s − 1.42·18-s − 0.689·19-s − 2.17·20-s − 0.304·21-s − 1.96·22-s − 0.232·23-s − 0.251·24-s + 1.10·25-s + 0.438·26-s + 0.599·27-s + 1.44·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 0.546T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 17 | \( 1 + 4.63T + 17T^{2} \) |
| 19 | \( 1 + 3.00T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 0.834T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 1.11T + 53T^{2} \) |
| 59 | \( 1 + 5.80T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 1.53T + 67T^{2} \) |
| 71 | \( 1 - 6.31T + 71T^{2} \) |
| 73 | \( 1 + 1.67T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 0.710T + 89T^{2} \) |
| 97 | \( 1 + 0.857T + 97T^{2} \) |
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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
−8.921130176006100758617766616299, −8.144303863778397421727212763904, −7.55079867797148796596516206660, −6.45924960004473462065880640739, −5.63378228075922103479706633732, −4.72412758014202689795954786644, −4.32934456001060977942434775700, −3.23752721873891802674818626559, −2.31431827310670143982184626632, 0,
2.31431827310670143982184626632, 3.23752721873891802674818626559, 4.32934456001060977942434775700, 4.72412758014202689795954786644, 5.63378228075922103479706633732, 6.45924960004473462065880640739, 7.55079867797148796596516206660, 8.144303863778397421727212763904, 8.921130176006100758617766616299
