| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 2·13-s − 4·14-s + 16-s − 6·17-s + 4·19-s − 20-s + 25-s − 2·26-s + 4·28-s − 6·29-s + 8·31-s − 32-s + 6·34-s − 4·35-s − 2·37-s − 4·38-s + 40-s + 6·41-s + 9·49-s − 50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.755·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 9/7·49-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.54499511285358, −13.05535819886198, −12.35200413986054, −11.75531168522060, −11.56204766454615, −11.10784467551665, −10.72311055619424, −10.27254422082945, −9.515966055068688, −9.107485239731250, −8.628693137236943, −8.122463575073473, −7.910318940446368, −7.195860218916557, −6.914912812510874, −6.186198265664266, −5.619538812345373, −5.057351700823260, −4.510363588884457, −4.026758692435959, −3.413227756628567, −2.521793810772198, −2.172380675866272, −1.352974234566935, −0.9422891903481882, 0,
0.9422891903481882, 1.352974234566935, 2.172380675866272, 2.521793810772198, 3.413227756628567, 4.026758692435959, 4.510363588884457, 5.057351700823260, 5.619538812345373, 6.186198265664266, 6.914912812510874, 7.195860218916557, 7.910318940446368, 8.122463575073473, 8.628693137236943, 9.107485239731250, 9.515966055068688, 10.27254422082945, 10.72311055619424, 11.10784467551665, 11.56204766454615, 11.75531168522060, 12.35200413986054, 13.05535819886198, 13.54499511285358
