| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 4·11-s + 13-s + 16-s − 2·17-s − 4·19-s − 2·20-s + 4·22-s − 4·23-s − 25-s − 26-s + 4·31-s − 32-s + 2·34-s + 4·37-s + 4·38-s + 2·40-s + 8·43-s − 4·44-s + 4·46-s + 6·47-s + 50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s − 0.834·23-s − 1/5·25-s − 0.196·26-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s + 0.648·38-s + 0.316·40-s + 1.21·43-s − 0.603·44-s + 0.589·46-s + 0.875·47-s + 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3966892327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3966892327\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−16.28821285003759, −15.92667639229676, −15.41583693154803, −15.06921589309269, −14.22111264032214, −13.53038440339133, −12.95280831745519, −12.29744173275515, −11.78781351775501, −11.14488576582273, −10.54799047392009, −10.25082504243663, −9.310749332187231, −8.733961492092074, −8.134109927877323, −7.622327741396790, −7.216941875941371, −6.066955202910740, −5.905286229982612, −4.538384497372288, −4.294958411394631, −3.158741013714261, −2.542741061864032, −1.599709997743728, −0.3157434039879697,
0.3157434039879697, 1.599709997743728, 2.542741061864032, 3.158741013714261, 4.294958411394631, 4.538384497372288, 5.905286229982612, 6.066955202910740, 7.216941875941371, 7.622327741396790, 8.134109927877323, 8.733961492092074, 9.310749332187231, 10.25082504243663, 10.54799047392009, 11.14488576582273, 11.78781351775501, 12.29744173275515, 12.95280831745519, 13.53038440339133, 14.22111264032214, 15.06921589309269, 15.41583693154803, 15.92667639229676, 16.28821285003759
