| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s + 13-s + 2·15-s + 4·17-s − 6·19-s + 6·23-s − 25-s − 27-s − 10·29-s − 4·31-s + 4·33-s − 4·37-s − 39-s − 2·41-s + 4·43-s − 2·45-s + 8·47-s − 4·51-s + 6·53-s + 8·55-s + 6·57-s − 6·59-s − 6·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s − 0.657·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 0.560·51-s + 0.824·53-s + 1.07·55-s + 0.794·57-s − 0.781·59-s − 0.768·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.03510795461333, −12.83742306732847, −12.52818870337851, −11.91143076010003, −11.38569837269935, −11.07096235546248, −10.65207682769837, −10.25010296023830, −9.748032660199532, −8.940803723746445, −8.762059451607576, −8.081266742611624, −7.584996673555554, −7.255436472328697, −6.915316534406234, −5.934793122753895, −5.725275783641887, −5.291688341735103, −4.572407313643578, −4.179436285555965, −3.595123033739523, −3.105609970799023, −2.407338023194954, −1.739996487773446, −1.036506658713868, 0, 0,
1.036506658713868, 1.739996487773446, 2.407338023194954, 3.105609970799023, 3.595123033739523, 4.179436285555965, 4.572407313643578, 5.291688341735103, 5.725275783641887, 5.934793122753895, 6.915316534406234, 7.255436472328697, 7.584996673555554, 8.081266742611624, 8.762059451607576, 8.940803723746445, 9.748032660199532, 10.25010296023830, 10.65207682769837, 11.07096235546248, 11.38569837269935, 11.91143076010003, 12.52818870337851, 12.83742306732847, 13.03510795461333
