| L(s) = 1 | − 2·3-s − 2·5-s − 4·7-s + 9-s − 2·11-s + 2·13-s + 4·15-s + 2·17-s + 2·19-s + 8·21-s − 25-s + 4·27-s − 6·29-s + 4·33-s + 8·35-s − 10·37-s − 4·39-s − 6·41-s + 6·43-s − 2·45-s + 8·47-s + 9·49-s − 4·51-s + 6·53-s + 4·55-s − 4·57-s − 14·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s + 0.458·19-s + 1.74·21-s − 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.696·33-s + 1.35·35-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−14.36232490959404, −13.77746573532842, −13.38484119993041, −12.66182810591663, −12.41896669104856, −11.93807819610073, −11.48624405392854, −10.88037172691711, −10.50693315556252, −10.02183017439700, −9.394478430462438, −8.853736503442573, −8.263227272964122, −7.476338581565131, −7.233707865503066, −6.516993145621589, −6.066992892099055, −5.491866799759207, −5.172030471926290, −4.224341675243469, −3.688928433417742, −3.239410648196120, −2.563063740723578, −1.461990157237688, −0.5285705204077028, 0,
0.5285705204077028, 1.461990157237688, 2.563063740723578, 3.239410648196120, 3.688928433417742, 4.224341675243469, 5.172030471926290, 5.491866799759207, 6.066992892099055, 6.516993145621589, 7.233707865503066, 7.476338581565131, 8.263227272964122, 8.853736503442573, 9.394478430462438, 10.02183017439700, 10.50693315556252, 10.88037172691711, 11.48624405392854, 11.93807819610073, 12.41896669104856, 12.66182810591663, 13.38484119993041, 13.77746573532842, 14.36232490959404
