| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s − 11-s + 2·13-s + 2·15-s − 2·17-s + 6·19-s − 4·21-s + 6·23-s − 25-s − 27-s + 2·29-s + 10·31-s + 33-s − 8·35-s + 37-s − 2·39-s − 10·41-s − 10·43-s − 2·45-s + 4·47-s + 9·49-s + 2·51-s − 10·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s + 1.37·19-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.174·33-s − 1.35·35-s + 0.164·37-s − 0.320·39-s − 1.56·41-s − 1.52·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914544797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914544797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.99122679706071, −13.45569597514315, −13.18949029351477, −12.21761996873961, −11.90720269059718, −11.51426686943175, −11.23242138317588, −10.63781362192979, −10.19110877093043, −9.514723849808920, −8.803834382053347, −8.275534777509799, −7.994888828889003, −7.409369685198743, −6.887745334095117, −6.329541959660146, −5.496011365037331, −5.112378445487104, −4.532176713761435, −4.262251763550578, −3.232915802179556, −2.913532664804390, −1.709254681818945, −1.316607004935142, −0.5065064241797911,
0.5065064241797911, 1.316607004935142, 1.709254681818945, 2.913532664804390, 3.232915802179556, 4.262251763550578, 4.532176713761435, 5.112378445487104, 5.496011365037331, 6.329541959660146, 6.887745334095117, 7.409369685198743, 7.994888828889003, 8.275534777509799, 8.803834382053347, 9.514723849808920, 10.19110877093043, 10.63781362192979, 11.23242138317588, 11.51426686943175, 11.90720269059718, 12.21761996873961, 13.18949029351477, 13.45569597514315, 13.99122679706071
