| L(s) = 1 | + 3-s + 2·7-s − 2·9-s − 3·11-s − 4·13-s + 4·19-s + 2·21-s + 9·23-s − 5·25-s − 5·27-s + 6·29-s + 31-s − 3·33-s − 5·37-s − 4·39-s + 12·41-s + 10·43-s − 3·47-s − 3·49-s − 6·53-s + 4·57-s + 3·59-s − 11·61-s − 4·63-s + 7·67-s + 9·69-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s + 0.917·19-s + 0.436·21-s + 1.87·23-s − 25-s − 0.962·27-s + 1.11·29-s + 0.179·31-s − 0.522·33-s − 0.821·37-s − 0.640·39-s + 1.87·41-s + 1.52·43-s − 0.437·47-s − 3/7·49-s − 0.824·53-s + 0.529·57-s + 0.390·59-s − 1.40·61-s − 0.503·63-s + 0.855·67-s + 1.08·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| 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
−13.73003096369317, −13.21046122722264, −12.64361551402343, −12.27538346827882, −11.63723489754888, −11.23009932433162, −10.84654443067055, −10.23717670767146, −9.719306992609394, −9.160394288706584, −8.895231184996724, −8.151914704425008, −7.756368825570716, −7.498250310415127, −6.922769913054690, −6.095169218488715, −5.618573639564841, −4.939870400983012, −4.848615616958284, −4.024528108031976, −3.240165028143073, −2.694446487247098, −2.510812447792436, −1.614410729212296, −0.8752058051357970, 0,
0.8752058051357970, 1.614410729212296, 2.510812447792436, 2.694446487247098, 3.240165028143073, 4.024528108031976, 4.848615616958284, 4.939870400983012, 5.618573639564841, 6.095169218488715, 6.922769913054690, 7.498250310415127, 7.756368825570716, 8.151914704425008, 8.895231184996724, 9.160394288706584, 9.719306992609394, 10.23717670767146, 10.84654443067055, 11.23009932433162, 11.63723489754888, 12.27538346827882, 12.64361551402343, 13.21046122722264, 13.73003096369317
