| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·13-s − 2·17-s − 6·19-s − 2·21-s − 5·25-s + 4·27-s − 2·37-s − 8·39-s − 10·41-s + 8·43-s − 8·47-s + 49-s + 4·51-s + 6·53-s + 12·57-s + 10·59-s + 12·61-s + 63-s − 12·67-s + 8·71-s − 14·73-s + 10·75-s − 11·81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.436·21-s − 25-s + 0.769·27-s − 0.328·37-s − 1.28·39-s − 1.56·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 1.58·57-s + 1.30·59-s + 1.53·61-s + 0.125·63-s − 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.15·75-s − 1.22·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.76959083491552, −14.49530565094828, −13.61911615093485, −13.30506931416188, −12.80171235050532, −12.10194952261605, −11.67582318356076, −11.26672465289566, −10.82077039964515, −10.30859234647449, −9.880461838998947, −8.811271660103754, −8.681160981520491, −8.066596073613741, −7.285564167549367, −6.634042545765372, −6.263173117133949, −5.714662700757663, −5.203822188646167, −4.507289658947354, −4.005270983695673, −3.323770913066300, −2.302030064657856, −1.716058284109685, −0.8082170297935037, 0,
0.8082170297935037, 1.716058284109685, 2.302030064657856, 3.323770913066300, 4.005270983695673, 4.507289658947354, 5.203822188646167, 5.714662700757663, 6.263173117133949, 6.634042545765372, 7.285564167549367, 8.066596073613741, 8.681160981520491, 8.811271660103754, 9.880461838998947, 10.30859234647449, 10.82077039964515, 11.26672465289566, 11.67582318356076, 12.10194952261605, 12.80171235050532, 13.30506931416188, 13.61911615093485, 14.49530565094828, 14.76959083491552
