| L(s) = 1 | − 1.93·2-s − 2.42·3-s + 1.75·4-s + 2.12·5-s + 4.69·6-s + 2.19·7-s + 0.478·8-s + 2.87·9-s − 4.11·10-s − 1.62·11-s − 4.25·12-s + 13-s − 4.24·14-s − 5.14·15-s − 4.43·16-s − 4.84·17-s − 5.57·18-s − 1.41·19-s + 3.72·20-s − 5.31·21-s + 3.14·22-s − 1.24·23-s − 1.16·24-s − 0.493·25-s − 1.93·26-s + 0.292·27-s + 3.83·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 1.39·3-s + 0.876·4-s + 0.949·5-s + 1.91·6-s + 0.827·7-s + 0.169·8-s + 0.959·9-s − 1.30·10-s − 0.489·11-s − 1.22·12-s + 0.277·13-s − 1.13·14-s − 1.32·15-s − 1.10·16-s − 1.17·17-s − 1.31·18-s − 0.325·19-s + 0.832·20-s − 1.15·21-s + 0.670·22-s − 0.259·23-s − 0.236·24-s − 0.0986·25-s − 0.379·26-s + 0.0562·27-s + 0.725·28-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 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 + 9.80T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.07T + 61T^{2} \) |
| 67 | \( 1 - 0.744T + 67T^{2} \) |
| 71 | \( 1 + 9.69T + 71T^{2} \) |
| 73 | \( 1 - 9.67T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−9.327272157611393494834017137303, −8.487018557036810580528889657588, −7.72752373210601443764372199634, −6.70088162145113378344629476837, −6.07183940085155263285339003813, −5.14254990113283000446658261913, −4.41380450754984119829777321245, −2.28215021970381713714020243832, −1.36885014731363553265913555146, 0,
1.36885014731363553265913555146, 2.28215021970381713714020243832, 4.41380450754984119829777321245, 5.14254990113283000446658261913, 6.07183940085155263285339003813, 6.70088162145113378344629476837, 7.72752373210601443764372199634, 8.487018557036810580528889657588, 9.327272157611393494834017137303
