| L(s) = 1 | − 2-s − 4-s − 5-s − 2·7-s + 3·8-s + 10-s − 11-s + 7·13-s + 2·14-s − 16-s − 17-s + 6·19-s + 20-s + 22-s − 8·23-s − 4·25-s − 7·26-s + 2·28-s − 3·29-s − 2·31-s − 5·32-s + 34-s + 2·35-s − 3·37-s − 6·38-s − 3·40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s + 0.316·10-s − 0.301·11-s + 1.94·13-s + 0.534·14-s − 1/4·16-s − 0.242·17-s + 1.37·19-s + 0.223·20-s + 0.213·22-s − 1.66·23-s − 4/5·25-s − 1.37·26-s + 0.377·28-s − 0.557·29-s − 0.359·31-s − 0.883·32-s + 0.171·34-s + 0.338·35-s − 0.493·37-s − 0.973·38-s − 0.474·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.688678298787394232032584366111, −8.841204386443289209009120453460, −8.158491432152574935865930788879, −7.42914874671437718782079244253, −6.27827312125011644540569752382, −5.39744005634419493992839888034, −4.02715200249010457921191571121, −3.43365430050361198222708919283, −1.53865115269998686381095233957, 0,
1.53865115269998686381095233957, 3.43365430050361198222708919283, 4.02715200249010457921191571121, 5.39744005634419493992839888034, 6.27827312125011644540569752382, 7.42914874671437718782079244253, 8.158491432152574935865930788879, 8.841204386443289209009120453460, 9.688678298787394232032584366111
