| L(s) = 1 | + 3-s + 5-s + 9-s + 15-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 2·41-s − 4·43-s + 45-s − 7·49-s + 2·51-s + 6·53-s − 4·57-s − 2·61-s + 8·67-s − 4·69-s − 6·73-s + 75-s + 4·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.256·61-s + 0.977·67-s − 0.481·69-s − 0.702·73-s + 0.115·75-s + 0.450·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.36889030188838, −13.65565925963003, −13.29482660406583, −12.86856791243216, −12.32565592024860, −11.76624682756786, −11.25818331348163, −10.64848741660731, −10.08469652282787, −9.762074352989412, −9.261466086680797, −8.601909948785004, −8.194887524372763, −7.728870873610708, −7.112849666345933, −6.441943862994995, −6.075456167161358, −5.435318063106905, −4.767005696842333, −4.166273207819739, −3.678358053122505, −2.891829258888078, −2.386814308111070, −1.754158038611956, −1.045066424038222, 0,
1.045066424038222, 1.754158038611956, 2.386814308111070, 2.891829258888078, 3.678358053122505, 4.166273207819739, 4.767005696842333, 5.435318063106905, 6.075456167161358, 6.441943862994995, 7.112849666345933, 7.728870873610708, 8.194887524372763, 8.601909948785004, 9.261466086680797, 9.762074352989412, 10.08469652282787, 10.64848741660731, 11.25818331348163, 11.76624682756786, 12.32565592024860, 12.86856791243216, 13.29482660406583, 13.65565925963003, 14.36889030188838
