| L(s) = 1 | + 5-s − 2·7-s − 3·9-s − 6·11-s + 2·13-s − 6·17-s + 2·19-s − 6·23-s + 25-s + 6·29-s − 4·31-s − 2·35-s + 6·37-s − 2·41-s − 4·43-s − 3·45-s + 10·47-s − 3·49-s + 2·53-s − 6·55-s − 10·59-s − 10·61-s + 6·63-s + 2·65-s + 4·67-s − 16·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.45·47-s − 3/7·49-s + 0.274·53-s − 0.809·55-s − 1.30·59-s − 1.28·61-s + 0.755·63-s + 0.248·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27119522394318848101695814883, −9.240805137182251408961722561285, −8.459530406419082967845984524867, −7.57454589175301650317308735633, −6.31867615323075833576541028992, −5.74481417491481656913857127073, −4.64601077885432605050562198244, −3.15828578541213785461358104860, −2.31650484656215274723281292429, 0,
2.31650484656215274723281292429, 3.15828578541213785461358104860, 4.64601077885432605050562198244, 5.74481417491481656913857127073, 6.31867615323075833576541028992, 7.57454589175301650317308735633, 8.459530406419082967845984524867, 9.240805137182251408961722561285, 10.27119522394318848101695814883
