| L(s) = 1 | − 3-s + 4·7-s + 9-s − 11-s − 13-s − 2·17-s + 4·19-s − 4·21-s − 27-s − 4·29-s − 2·31-s + 33-s − 2·37-s + 39-s − 6·41-s − 6·43-s − 4·47-s + 9·49-s + 2·51-s + 6·53-s − 4·57-s + 4·59-s + 10·61-s + 4·63-s − 12·67-s − 4·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.872·21-s − 0.192·27-s − 0.742·29-s − 0.359·31-s + 0.174·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.914·43-s − 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.503·63-s − 1.46·67-s − 0.474·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.43550636972031, −13.03796457485062, −12.38356337918972, −11.81620554410852, −11.61799035526753, −11.15101647177885, −10.74290530535150, −10.11782182575652, −9.835250093707417, −9.060367412008804, −8.629268329879135, −8.125831223083761, −7.647144953206283, −7.130406100032865, −6.799212749689258, −5.953074181636470, −5.482532333967133, −5.057837908039234, −4.715384632223190, −4.062244752385263, −3.473655801462577, −2.731734134852999, −1.930793861223095, −1.630257722649355, −0.8343069024011901, 0,
0.8343069024011901, 1.630257722649355, 1.930793861223095, 2.731734134852999, 3.473655801462577, 4.062244752385263, 4.715384632223190, 5.057837908039234, 5.482532333967133, 5.953074181636470, 6.799212749689258, 7.130406100032865, 7.647144953206283, 8.125831223083761, 8.629268329879135, 9.060367412008804, 9.835250093707417, 10.11782182575652, 10.74290530535150, 11.15101647177885, 11.61799035526753, 11.81620554410852, 12.38356337918972, 13.03796457485062, 13.43550636972031
