| L(s) = 1 | − 3-s − 2·5-s + 2·7-s − 2·9-s + 2·15-s − 7·17-s + 2·19-s − 2·21-s + 3·23-s − 25-s + 5·27-s − 9·29-s − 6·31-s − 4·35-s − 6·37-s − 6·41-s − 5·43-s + 4·45-s + 6·47-s − 3·49-s + 7·51-s + 11·53-s − 2·57-s − 8·59-s + 5·61-s − 4·63-s − 14·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s − 2/3·9-s + 0.516·15-s − 1.69·17-s + 0.458·19-s − 0.436·21-s + 0.625·23-s − 1/5·25-s + 0.962·27-s − 1.67·29-s − 1.07·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s − 0.762·43-s + 0.596·45-s + 0.875·47-s − 3/7·49-s + 0.980·51-s + 1.51·53-s − 0.264·57-s − 1.04·59-s + 0.640·61-s − 0.503·63-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.16027358203272, −12.38555765846874, −12.02763477445040, −11.63122267376566, −11.27919967999547, −10.90312094517158, −10.64785013930415, −9.961590550522317, −9.212247479157045, −8.825724555985512, −8.631160189053024, −7.945895131500311, −7.448768767192844, −7.135856330554196, −6.614463330135140, −5.937341742089430, −5.560723985601628, −4.947225885047507, −4.702112726420663, −4.012536902936642, −3.524398326950824, −3.063562384296294, −2.112156301114202, −1.883206845536653, −0.9734362177510992, 0, 0,
0.9734362177510992, 1.883206845536653, 2.112156301114202, 3.063562384296294, 3.524398326950824, 4.012536902936642, 4.702112726420663, 4.947225885047507, 5.560723985601628, 5.937341742089430, 6.614463330135140, 7.135856330554196, 7.448768767192844, 7.945895131500311, 8.631160189053024, 8.825724555985512, 9.212247479157045, 9.961590550522317, 10.64785013930415, 10.90312094517158, 11.27919967999547, 11.63122267376566, 12.02763477445040, 12.38555765846874, 13.16027358203272
