| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s − 11-s + 2·15-s − 3·17-s + 5·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s + 2·31-s − 2·33-s + 2·35-s − 5·37-s − 9·41-s + 43-s + 45-s − 3·47-s − 3·49-s − 6·51-s − 55-s + 10·57-s − 12·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s − 0.727·17-s + 1.14·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s + 0.338·35-s − 0.821·37-s − 1.40·41-s + 0.152·43-s + 0.149·45-s − 0.437·47-s − 3/7·49-s − 0.840·51-s − 0.134·55-s + 1.32·57-s − 1.56·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
| 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
−13.65159143113804, −13.36261346205287, −12.75515513946526, −12.26176031842179, −11.54504506817780, −11.17776645442811, −10.83843213335171, −10.03773521665810, −9.696686282852094, −9.097390259603736, −8.867679628103982, −8.184363933916484, −7.963728042283433, −7.261183051902639, −6.920008971796381, −6.243677081336584, −5.479927956602919, −5.091639085825570, −4.694980898507088, −3.791807652933643, −3.363292431148068, −2.841029593017837, −2.190930573662104, −1.735964668646830, −1.081898142370463, 0,
1.081898142370463, 1.735964668646830, 2.190930573662104, 2.841029593017837, 3.363292431148068, 3.791807652933643, 4.694980898507088, 5.091639085825570, 5.479927956602919, 6.243677081336584, 6.920008971796381, 7.261183051902639, 7.963728042283433, 8.184363933916484, 8.867679628103982, 9.097390259603736, 9.696686282852094, 10.03773521665810, 10.83843213335171, 11.17776645442811, 11.54504506817780, 12.26176031842179, 12.75515513946526, 13.36261346205287, 13.65159143113804
