| L(s) = 1 | − 2.47·2-s + 2.80·3-s + 4.14·4-s − 2.79·5-s − 6.96·6-s + 0.439·7-s − 5.32·8-s + 4.88·9-s + 6.93·10-s − 2.41·11-s + 11.6·12-s + 13-s − 1.09·14-s − 7.84·15-s + 4.90·16-s − 6.86·17-s − 12.1·18-s + 0.555·19-s − 11.5·20-s + 1.23·21-s + 5.98·22-s + 0.601·23-s − 14.9·24-s + 2.81·25-s − 2.47·26-s + 5.29·27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 1.62·3-s + 2.07·4-s − 1.24·5-s − 2.84·6-s + 0.166·7-s − 1.88·8-s + 1.62·9-s + 2.19·10-s − 0.728·11-s + 3.36·12-s + 0.277·13-s − 0.291·14-s − 2.02·15-s + 1.22·16-s − 1.66·17-s − 2.85·18-s + 0.127·19-s − 2.59·20-s + 0.269·21-s + 1.27·22-s + 0.125·23-s − 3.05·24-s + 0.562·25-s − 0.486·26-s + 1.01·27-s + 0.344·28-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 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.439T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 19 | \( 1 - 0.555T + 19T^{2} \) |
| 23 | \( 1 - 0.601T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 0.0596T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 - 0.473T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.817T + 53T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 0.957T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 8.97T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 9.17T + 97T^{2} \) |
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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
−8.998125452575921824863718503815, −8.307263912529668949400525759458, −8.065960229711539881099982984309, −7.28524247494873585187754989614, −6.61420599736490255836387576290, −4.68761404845487239128469656193, −3.58354690510020497097516077512, −2.67262587616663060225924270628, −1.71762308867140867244522852433, 0,
1.71762308867140867244522852433, 2.67262587616663060225924270628, 3.58354690510020497097516077512, 4.68761404845487239128469656193, 6.61420599736490255836387576290, 7.28524247494873585187754989614, 8.065960229711539881099982984309, 8.307263912529668949400525759458, 8.998125452575921824863718503815
