| L(s) = 1 | + 3-s − 2·4-s − 3·5-s − 4·7-s − 2·9-s + 11-s − 2·12-s + 2·13-s − 3·15-s + 4·16-s + 19-s + 6·20-s − 4·21-s + 3·23-s + 4·25-s − 5·27-s + 8·28-s − 6·29-s − 7·31-s + 33-s + 12·35-s + 4·36-s − 7·37-s + 2·39-s − 10·43-s − 2·44-s + 6·45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s − 1.51·7-s − 2/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 0.774·15-s + 16-s + 0.229·19-s + 1.34·20-s − 0.872·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 1.51·28-s − 1.11·29-s − 1.25·31-s + 0.174·33-s + 2.02·35-s + 2/3·36-s − 1.15·37-s + 0.320·39-s − 1.52·43-s − 0.301·44-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.05917203220889978964010728308, −10.96102192801302376001650382810, −9.603034074754024658706636273886, −8.916975147724513941001822680471, −8.090908771111864109123836483338, −6.92292199886679346603140457685, −5.46804944393504386563865646601, −3.77762904601741073110269049747, −3.37641666123212985170587332393, 0,
3.37641666123212985170587332393, 3.77762904601741073110269049747, 5.46804944393504386563865646601, 6.92292199886679346603140457685, 8.090908771111864109123836483338, 8.916975147724513941001822680471, 9.603034074754024658706636273886, 10.96102192801302376001650382810, 12.05917203220889978964010728308
