| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 4·11-s − 2·13-s − 2·15-s + 2·17-s − 19-s − 21-s + 23-s − 25-s + 27-s + 2·29-s + 4·33-s + 2·35-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s − 2·45-s + 49-s + 2·51-s + 6·53-s − 8·55-s − 57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.485·17-s − 0.229·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.132·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 146832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 146832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550647622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550647622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−13.39493603437202, −12.84236584018482, −12.34033618300365, −11.96631275735288, −11.54758921493566, −11.13481576686157, −10.22703759803884, −10.10911113620046, −9.421795948242498, −8.992262703069276, −8.556605470144066, −7.953173449766788, −7.591071014488141, −7.051011470192710, −6.472672119616973, −6.206476418811412, −5.147276282329147, −4.903378412411059, −4.034752028651426, −3.695068283589649, −3.375473708729500, −2.515107296352807, −2.002330013990620, −1.129827381300056, −0.5015141517197985,
0.5015141517197985, 1.129827381300056, 2.002330013990620, 2.515107296352807, 3.375473708729500, 3.695068283589649, 4.034752028651426, 4.903378412411059, 5.147276282329147, 6.206476418811412, 6.472672119616973, 7.051011470192710, 7.591071014488141, 7.953173449766788, 8.556605470144066, 8.992262703069276, 9.421795948242498, 10.10911113620046, 10.22703759803884, 11.13481576686157, 11.54758921493566, 11.96631275735288, 12.34033618300365, 12.84236584018482, 13.39493603437202
