| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s + 10-s + 13-s − 16-s + 5·17-s + 6·19-s − 20-s − 2·23-s − 4·25-s + 26-s + 9·29-s + 2·31-s + 5·32-s + 5·34-s − 3·37-s + 6·38-s − 3·40-s + 5·41-s − 2·46-s + 2·47-s − 4·50-s − 52-s − 9·53-s + 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s + 0.277·13-s − 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.223·20-s − 0.417·23-s − 4/5·25-s + 0.196·26-s + 1.67·29-s + 0.359·31-s + 0.883·32-s + 0.857·34-s − 0.493·37-s + 0.973·38-s − 0.474·40-s + 0.780·41-s − 0.294·46-s + 0.291·47-s − 0.565·50-s − 0.138·52-s − 1.23·53-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.560599731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560599731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.26511914762322, −14.00820660375583, −13.52665873213066, −13.02671705278764, −12.43513114363835, −11.92360679525424, −11.72175521858312, −10.87912893448001, −10.12507799038472, −9.804994676351761, −9.457136703734291, −8.617930169046335, −8.285752497541416, −7.591992759278952, −7.032021291390548, −6.060441516418080, −5.978449233899508, −5.269382670260231, −4.794978416145487, −4.148882278939195, −3.420906103751609, −3.077666811959225, −2.267704896327643, −1.273565800950637, −0.6473354394963431,
0.6473354394963431, 1.273565800950637, 2.267704896327643, 3.077666811959225, 3.420906103751609, 4.148882278939195, 4.794978416145487, 5.269382670260231, 5.978449233899508, 6.060441516418080, 7.032021291390548, 7.591992759278952, 8.285752497541416, 8.617930169046335, 9.457136703734291, 9.804994676351761, 10.12507799038472, 10.87912893448001, 11.72175521858312, 11.92360679525424, 12.43513114363835, 13.02671705278764, 13.52665873213066, 14.00820660375583, 14.26511914762322
