| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 2·13-s − 15-s + 2·17-s − 4·19-s − 21-s + 25-s − 27-s + 2·29-s − 8·31-s + 35-s − 6·37-s − 2·39-s − 10·41-s + 45-s + 49-s − 2·51-s + 53-s + 4·57-s − 2·61-s + 63-s + 2·65-s + 12·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.169·35-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.149·45-s + 1/7·49-s − 0.280·51-s + 0.137·53-s + 0.529·57-s − 0.256·61-s + 0.125·63-s + 0.248·65-s + 1.46·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 53 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.91518906478586, −14.29316094838481, −14.00572753740714, −13.19711829150548, −12.96000111157696, −12.25416639974809, −11.88152023171739, −11.17490967800935, −10.79082226750384, −10.31675181922818, −9.794243084215545, −9.102405159757223, −8.603380373647017, −8.075269895984879, −7.398827780177900, −6.707572085124288, −6.391148109470172, −5.605531626751880, −5.218946644862882, −4.657864963030026, −3.782002979822729, −3.403669936875793, −2.286069004982325, −1.777160298539208, −1.006217251874665, 0,
1.006217251874665, 1.777160298539208, 2.286069004982325, 3.403669936875793, 3.782002979822729, 4.657864963030026, 5.218946644862882, 5.605531626751880, 6.391148109470172, 6.707572085124288, 7.398827780177900, 8.075269895984879, 8.603380373647017, 9.102405159757223, 9.794243084215545, 10.31675181922818, 10.79082226750384, 11.17490967800935, 11.88152023171739, 12.25416639974809, 12.96000111157696, 13.19711829150548, 14.00572753740714, 14.29316094838481, 14.91518906478586
