| L(s) = 1 | + 5-s + 7-s − 4·11-s − 2·13-s − 3·17-s + 8·19-s − 6·23-s − 4·25-s + 4·29-s − 6·31-s + 35-s − 3·37-s − 41-s − 11·43-s + 9·47-s + 49-s − 6·53-s − 4·55-s − 15·59-s + 4·61-s − 2·65-s + 8·67-s − 12·71-s + 6·73-s − 4·77-s + 79-s − 9·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.727·17-s + 1.83·19-s − 1.25·23-s − 4/5·25-s + 0.742·29-s − 1.07·31-s + 0.169·35-s − 0.493·37-s − 0.156·41-s − 1.67·43-s + 1.31·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s − 1.95·59-s + 0.512·61-s − 0.248·65-s + 0.977·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s + 0.112·79-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.202528816763435832367665093642, −7.67144753547016948945764794343, −6.95780617830712445836046912571, −5.88613407491592695110791197579, −5.30625936839298596651319785349, −4.60863294327089660070950693888, −3.46065070971004323061547085118, −2.50870406717853018462127950651, −1.64303547061960194320082119401, 0,
1.64303547061960194320082119401, 2.50870406717853018462127950651, 3.46065070971004323061547085118, 4.60863294327089660070950693888, 5.30625936839298596651319785349, 5.88613407491592695110791197579, 6.95780617830712445836046912571, 7.67144753547016948945764794343, 8.202528816763435832367665093642
