| L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 4·13-s − 4·19-s − 2·21-s − 4·23-s − 27-s + 29-s − 8·31-s − 33-s + 2·37-s − 4·39-s + 2·41-s − 2·43-s + 12·47-s − 3·49-s + 10·53-s + 4·57-s + 12·59-s − 14·61-s + 2·63-s + 4·69-s − 8·71-s − 4·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.917·19-s − 0.436·21-s − 0.834·23-s − 0.192·27-s + 0.185·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.640·39-s + 0.312·41-s − 0.304·43-s + 1.75·47-s − 3/7·49-s + 1.37·53-s + 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.251·63-s + 0.481·69-s − 0.949·71-s − 0.468·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.63310767404697, −12.10624837103456, −11.84203650534243, −11.22757529564002, −11.00902095364431, −10.48303316047596, −10.21638361616677, −9.520920428328366, −8.929567431766652, −8.673078447115864, −8.220769367990688, −7.484350731785080, −7.359982817670425, −6.586482587921374, −6.158735887843442, −5.760552683101944, −5.373914213973770, −4.622014038516451, −4.286719374315264, −3.793466805143395, −3.324428349192256, −2.375679237069143, −1.976723179135619, −1.364758531816583, −0.8000593664790446, 0,
0.8000593664790446, 1.364758531816583, 1.976723179135619, 2.375679237069143, 3.324428349192256, 3.793466805143395, 4.286719374315264, 4.622014038516451, 5.373914213973770, 5.760552683101944, 6.158735887843442, 6.586482587921374, 7.359982817670425, 7.484350731785080, 8.220769367990688, 8.673078447115864, 8.929567431766652, 9.520920428328366, 10.21638361616677, 10.48303316047596, 11.00902095364431, 11.22757529564002, 11.84203650534243, 12.10624837103456, 12.63310767404697
