| L(s) = 1 | + 7·5-s − 21·7-s − 6·11-s + 13·13-s + 115·17-s − 46·19-s − 144·23-s − 76·25-s + 162·29-s + 180·31-s − 147·35-s + 13·37-s − 192·41-s − 33·43-s − 383·47-s + 98·49-s − 288·53-s − 42·55-s − 442·59-s − 680·61-s + 91·65-s − 722·67-s + 207·71-s + 274·73-s + 126·77-s − 936·79-s + 1.20e3·83-s + ⋯ |
| L(s) = 1 | + 0.626·5-s − 1.13·7-s − 0.164·11-s + 0.277·13-s + 1.64·17-s − 0.555·19-s − 1.30·23-s − 0.607·25-s + 1.03·29-s + 1.04·31-s − 0.709·35-s + 0.0577·37-s − 0.731·41-s − 0.117·43-s − 1.18·47-s + 2/7·49-s − 0.746·53-s − 0.102·55-s − 0.975·59-s − 1.42·61-s + 0.173·65-s − 1.31·67-s + 0.346·71-s + 0.439·73-s + 0.186·77-s − 1.33·79-s + 1.59·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 115 T + p^{3} T^{2} \) |
| 19 | \( 1 + 46 T + p^{3} T^{2} \) |
| 23 | \( 1 + 144 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 13 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 + 33 T + p^{3} T^{2} \) |
| 47 | \( 1 + 383 T + p^{3} T^{2} \) |
| 53 | \( 1 + 288 T + p^{3} T^{2} \) |
| 59 | \( 1 + 442 T + p^{3} T^{2} \) |
| 61 | \( 1 + 680 T + p^{3} T^{2} \) |
| 67 | \( 1 + 722 T + p^{3} T^{2} \) |
| 71 | \( 1 - 207 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 + 936 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 966 T + p^{3} T^{2} \) |
| 97 | \( 1 + 138 T + p^{3} T^{2} \) |
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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
−9.548889311420418277493236959645, −8.386158252367839025477068558225, −7.64736570619234577189786973290, −6.29450302481893694352509617175, −6.13005931620180943039094551304, −4.89990356564291798349602829763, −3.65144629439520133174915207607, −2.80053279352450176990017830937, −1.48210673471517056331740968285, 0,
1.48210673471517056331740968285, 2.80053279352450176990017830937, 3.65144629439520133174915207607, 4.89990356564291798349602829763, 6.13005931620180943039094551304, 6.29450302481893694352509617175, 7.64736570619234577189786973290, 8.386158252367839025477068558225, 9.548889311420418277493236959645
