| L(s) = 1 | − 3-s + 2·5-s + 9-s − 2·11-s − 2·15-s − 6·17-s + 8·19-s − 6·23-s − 25-s − 27-s − 10·29-s + 4·31-s + 2·33-s − 6·37-s − 6·41-s + 4·43-s + 2·45-s + 8·47-s + 6·51-s + 2·53-s − 4·55-s − 8·57-s − 4·59-s + 8·61-s + 8·67-s + 6·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s − 1.45·17-s + 1.83·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 0.840·51-s + 0.274·53-s − 0.539·55-s − 1.05·57-s − 0.520·59-s + 1.02·61-s + 0.977·67-s + 0.722·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.93002179114811, −13.45657950589453, −13.21034035124907, −12.52022433014024, −11.99029658940554, −11.60115990235100, −11.01594786042014, −10.60525677018568, −10.03740931067433, −9.577577154174872, −9.265543284458775, −8.587750307826455, −7.865426070596379, −7.525773187690108, −6.766442210575797, −6.473148126412994, −5.635407004194131, −5.452622375696084, −5.001855638216670, −4.096748385972820, −3.721608121197062, −2.826105664735673, −2.115831190751205, −1.792497819381195, −0.8107278507501693, 0,
0.8107278507501693, 1.792497819381195, 2.115831190751205, 2.826105664735673, 3.721608121197062, 4.096748385972820, 5.001855638216670, 5.452622375696084, 5.635407004194131, 6.473148126412994, 6.766442210575797, 7.525773187690108, 7.865426070596379, 8.587750307826455, 9.265543284458775, 9.577577154174872, 10.03740931067433, 10.60525677018568, 11.01594786042014, 11.60115990235100, 11.99029658940554, 12.52022433014024, 13.21034035124907, 13.45657950589453, 13.93002179114811
