| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s + 6·17-s + 2·19-s + 4·21-s + 2·23-s − 5·25-s − 27-s + 10·29-s + 4·31-s + 33-s − 37-s − 2·41-s − 6·43-s + 9·49-s − 6·51-s + 2·53-s − 2·57-s − 6·59-s + 4·61-s − 4·63-s − 4·67-s − 2·69-s − 12·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.45·17-s + 0.458·19-s + 0.872·21-s + 0.417·23-s − 25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.174·33-s − 0.164·37-s − 0.312·41-s − 0.914·43-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.264·57-s − 0.781·59-s + 0.512·61-s − 0.503·63-s − 0.488·67-s − 0.240·69-s − 1.42·71-s + 1.17·73-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.08437449606561, −13.67907244257509, −13.32248474414162, −12.64907875410484, −12.24977391084263, −11.93661889270885, −11.40298445764670, −10.59426224382423, −10.16473115116405, −9.874476831427260, −9.477791986746536, −8.686322620506987, −8.172925880350228, −7.512302836223229, −7.060287358351129, −6.360566836846717, −6.153972728403819, −5.443958525442948, −4.990365761587624, −4.254271263945834, −3.524928815986906, −3.088553342471130, −2.561945893716086, −1.464582951810937, −0.8010888941525402, 0,
0.8010888941525402, 1.464582951810937, 2.561945893716086, 3.088553342471130, 3.524928815986906, 4.254271263945834, 4.990365761587624, 5.443958525442948, 6.153972728403819, 6.360566836846717, 7.060287358351129, 7.512302836223229, 8.172925880350228, 8.686322620506987, 9.477791986746536, 9.874476831427260, 10.16473115116405, 10.59426224382423, 11.40298445764670, 11.93661889270885, 12.24977391084263, 12.64907875410484, 13.32248474414162, 13.67907244257509, 14.08437449606561
