| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 6·11-s − 13-s − 2·14-s + 16-s + 5·19-s + 20-s − 6·22-s − 2·23-s + 25-s + 26-s + 2·28-s − 6·29-s − 2·31-s − 32-s + 2·35-s − 5·37-s − 5·38-s − 40-s − 3·41-s + 10·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.14·19-s + 0.223·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.338·35-s − 0.821·37-s − 0.811·38-s − 0.158·40-s − 0.468·41-s + 1.52·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.58019824138742, −12.19035884929921, −11.86243449234882, −11.34301403877737, −11.12940173311430, −10.41189923689562, −10.07320273346627, −9.482978181510800, −9.154235612452727, −8.835596026149665, −8.364518844004237, −7.545735494871610, −7.416206856438878, −6.941658618699489, −6.275856156907699, −5.874890560540076, −5.417801805503369, −4.818753491606103, −4.190805398312436, −3.701594333897081, −3.200569787961815, −2.373475239933007, −1.901581904617801, −1.328256998140138, −0.9868600340395555, 0,
0.9868600340395555, 1.328256998140138, 1.901581904617801, 2.373475239933007, 3.200569787961815, 3.701594333897081, 4.190805398312436, 4.818753491606103, 5.417801805503369, 5.874890560540076, 6.275856156907699, 6.941658618699489, 7.416206856438878, 7.545735494871610, 8.364518844004237, 8.835596026149665, 9.154235612452727, 9.482978181510800, 10.07320273346627, 10.41189923689562, 11.12940173311430, 11.34301403877737, 11.86243449234882, 12.19035884929921, 12.58019824138742
