| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s + 11-s − 2·15-s + 2·17-s − 4·19-s − 4·21-s − 25-s + 27-s − 6·29-s − 4·31-s + 33-s + 8·35-s + 2·37-s + 6·41-s − 4·43-s − 2·45-s + 8·47-s + 9·49-s + 2·51-s − 6·53-s − 2·55-s − 4·57-s − 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 1.35·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.269·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.63027130026993, −13.00770253359679, −12.78349570158356, −12.28993164875841, −11.94201009398436, −11.22068912369628, −10.80150687843808, −10.34933496944815, −9.635279625648795, −9.407967539079656, −8.964084325513909, −8.415494112640799, −7.732359585473594, −7.525792273813013, −6.947218130062083, −6.387303397172313, −5.947067102368017, −5.421548242376079, −4.470444634423696, −4.079252620546242, −3.691379518461225, −3.083082188933953, −2.728383814143117, −1.882645513508074, −1.194527675544469, 0, 0,
1.194527675544469, 1.882645513508074, 2.728383814143117, 3.083082188933953, 3.691379518461225, 4.079252620546242, 4.470444634423696, 5.421548242376079, 5.947067102368017, 6.387303397172313, 6.947218130062083, 7.525792273813013, 7.732359585473594, 8.415494112640799, 8.964084325513909, 9.407967539079656, 9.635279625648795, 10.34933496944815, 10.80150687843808, 11.22068912369628, 11.94201009398436, 12.28993164875841, 12.78349570158356, 13.00770253359679, 13.63027130026993
