| L(s) = 1 | − 3-s + 2·5-s + 9-s − 6·11-s − 2·13-s − 2·15-s − 19-s + 4·23-s − 25-s − 27-s + 6·29-s − 2·31-s + 6·33-s − 8·37-s + 2·39-s − 2·41-s − 12·43-s + 2·45-s − 2·47-s + 6·53-s − 12·55-s + 57-s − 4·59-s + 2·61-s − 4·65-s − 6·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.516·15-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s + 0.298·45-s − 0.291·47-s + 0.824·53-s − 1.61·55-s + 0.132·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-s − 0.733·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−13.28218307949366, −13.07815278987379, −12.47049573474461, −12.04955454249871, −11.54059823107100, −10.92631091897700, −10.40824984539471, −10.24017099265805, −9.821766038195222, −9.184428189062893, −8.601936543143272, −8.162175334210661, −7.554441664381722, −7.097791319294856, −6.544019015392751, −6.106118435468163, −5.374309787140518, −5.124255588383381, −4.859568643833397, −4.018487951149429, −3.175753328883022, −2.799615852320842, −2.054545639409636, −1.684152227020327, −0.6785265877897367, 0,
0.6785265877897367, 1.684152227020327, 2.054545639409636, 2.799615852320842, 3.175753328883022, 4.018487951149429, 4.859568643833397, 5.124255588383381, 5.374309787140518, 6.106118435468163, 6.544019015392751, 7.097791319294856, 7.554441664381722, 8.162175334210661, 8.601936543143272, 9.184428189062893, 9.821766038195222, 10.24017099265805, 10.40824984539471, 10.92631091897700, 11.54059823107100, 12.04955454249871, 12.47049573474461, 13.07815278987379, 13.28218307949366
