| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 2·13-s − 17-s − 4·21-s − 2·23-s + 4·27-s − 6·29-s − 8·31-s + 10·37-s − 4·39-s − 2·41-s + 2·43-s + 6·47-s − 3·49-s + 2·51-s + 2·53-s + 8·59-s − 2·61-s + 2·63-s − 2·67-s + 4·69-s + 8·71-s − 2·73-s − 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.242·17-s − 0.872·21-s − 0.417·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.640·39-s − 0.312·41-s + 0.304·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.274·53-s + 1.04·59-s − 0.256·61-s + 0.251·63-s − 0.244·67-s + 0.481·69-s + 0.949·71-s − 0.234·73-s − 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−15.67511090119334, −14.81132001591030, −14.68171806815042, −13.90635933118725, −13.33505420586032, −12.72572864029031, −12.31239551129244, −11.49360866845511, −11.28319708971285, −10.93335250188065, −10.27275573662873, −9.605131828000078, −8.958548454023538, −8.388104609497318, −7.736325109379403, −7.166888260313983, −6.509510194304795, −5.712599857704046, −5.640832254501317, −4.800529397136681, −4.207627662288105, −3.560673471165628, −2.532516420109014, −1.749008857743803, −0.9435516195299060, 0,
0.9435516195299060, 1.749008857743803, 2.532516420109014, 3.560673471165628, 4.207627662288105, 4.800529397136681, 5.640832254501317, 5.712599857704046, 6.509510194304795, 7.166888260313983, 7.736325109379403, 8.388104609497318, 8.958548454023538, 9.605131828000078, 10.27275573662873, 10.93335250188065, 11.28319708971285, 11.49360866845511, 12.31239551129244, 12.72572864029031, 13.33505420586032, 13.90635933118725, 14.68171806815042, 14.81132001591030, 15.67511090119334
