| L(s) = 1 | + 7-s + 4·11-s + 2·13-s − 2·17-s + 19-s − 6·23-s − 5·25-s − 2·29-s − 6·31-s + 4·37-s − 6·41-s + 4·43-s − 12·47-s + 49-s + 10·53-s − 14·61-s − 6·67-s − 4·71-s − 14·73-s + 4·77-s + 4·79-s − 6·83-s − 6·89-s + 2·91-s + 8·97-s + 4·101-s − 6·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 25-s − 0.371·29-s − 1.07·31-s + 0.657·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.37·53-s − 1.79·61-s − 0.733·67-s − 0.474·71-s − 1.63·73-s + 0.455·77-s + 0.450·79-s − 0.658·83-s − 0.635·89-s + 0.209·91-s + 0.812·97-s + 0.398·101-s − 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.42098508901367458326261994504, −6.58143623204682662048436908707, −6.03252470357396858218331170667, −5.40374989671548728597605999950, −4.33478634831561441409532521398, −3.97758134940367204102265567381, −3.13004792404111322618176818002, −1.94903610600674587926317182730, −1.41672532431965057381966713353, 0,
1.41672532431965057381966713353, 1.94903610600674587926317182730, 3.13004792404111322618176818002, 3.97758134940367204102265567381, 4.33478634831561441409532521398, 5.40374989671548728597605999950, 6.03252470357396858218331170667, 6.58143623204682662048436908707, 7.42098508901367458326261994504
