| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 4·13-s − 2·15-s − 2·17-s + 8·19-s + 4·21-s − 4·23-s − 25-s − 27-s + 6·29-s − 8·35-s + 37-s + 4·39-s − 6·41-s + 8·43-s + 2·45-s + 6·47-s + 9·49-s + 2·51-s − 8·57-s − 6·59-s − 8·61-s − 4·63-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 0.485·17-s + 1.83·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.35·35-s + 0.164·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s + 0.875·47-s + 9/7·49-s + 0.280·51-s − 1.05·57-s − 0.781·59-s − 1.02·61-s − 0.503·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.15091519071990, −12.75327831571126, −12.27446212797813, −11.92609430680855, −11.52244702263745, −10.74892738567468, −10.24755096856313, −9.926110356160691, −9.636297454279679, −9.202919815854732, −8.715177639434757, −7.799350369698858, −7.437532998856192, −6.944624848249982, −6.415440068279501, −5.981553310532819, −5.676933783901839, −4.988924639066430, −4.613052113901509, −3.795350819793941, −3.302084810090244, −2.632338839337187, −2.284762415408751, −1.417449837737102, −0.6953378202124864, 0,
0.6953378202124864, 1.417449837737102, 2.284762415408751, 2.632338839337187, 3.302084810090244, 3.795350819793941, 4.613052113901509, 4.988924639066430, 5.676933783901839, 5.981553310532819, 6.415440068279501, 6.944624848249982, 7.437532998856192, 7.799350369698858, 8.715177639434757, 9.202919815854732, 9.636297454279679, 9.926110356160691, 10.24755096856313, 10.74892738567468, 11.52244702263745, 11.92609430680855, 12.27446212797813, 12.75327831571126, 13.15091519071990
