| L(s) = 1 | − 3·9-s − 2·11-s − 2·13-s + 6·17-s + 8·19-s + 8·23-s − 5·25-s + 4·29-s − 8·31-s − 8·37-s + 6·41-s − 8·43-s − 8·47-s − 7·49-s + 10·53-s + 2·59-s + 10·61-s − 14·67-s − 71-s + 14·73-s + 9·81-s + 12·83-s + 6·89-s − 2·97-s + 6·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 9-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 1.66·23-s − 25-s + 0.742·29-s − 1.43·31-s − 1.31·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s − 49-s + 1.37·53-s + 0.260·59-s + 1.28·61-s − 1.71·67-s − 0.118·71-s + 1.63·73-s + 81-s + 1.31·83-s + 0.635·89-s − 0.203·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724630751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724630751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 71 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.95373928151980, −15.05638091186653, −14.72687687300093, −14.19578634912464, −13.61722090102447, −13.13026036104957, −12.38033408530822, −11.84356746537596, −11.50699146304870, −10.76229899477929, −10.18790181142589, −9.550853584479386, −9.151446223789885, −8.300627470193035, −7.815938221448323, −7.280698100754651, −6.648227000108450, −5.614438069563625, −5.323368796642002, −4.927618571017279, −3.557516040637573, −3.264205710361678, −2.551594744169543, −1.517792884389967, −0.5627325281046834,
0.5627325281046834, 1.517792884389967, 2.551594744169543, 3.264205710361678, 3.557516040637573, 4.927618571017279, 5.323368796642002, 5.614438069563625, 6.648227000108450, 7.280698100754651, 7.815938221448323, 8.300627470193035, 9.151446223789885, 9.550853584479386, 10.18790181142589, 10.76229899477929, 11.50699146304870, 11.84356746537596, 12.38033408530822, 13.13026036104957, 13.61722090102447, 14.19578634912464, 14.72687687300093, 15.05638091186653, 15.95373928151980
