| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 4·11-s + 2·13-s − 15-s − 19-s + 2·21-s + 8·23-s + 25-s − 27-s + 2·29-s + 10·31-s − 4·33-s − 2·35-s − 6·37-s − 2·39-s + 12·41-s − 8·43-s + 45-s + 8·47-s − 3·49-s + 6·53-s + 4·55-s + 57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.229·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.79·31-s − 0.696·33-s − 0.338·35-s − 0.986·37-s − 0.320·39-s + 1.87·41-s − 1.21·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.539·55-s + 0.132·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.316932796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316932796\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 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
−15.75780241604222, −15.36425044428285, −14.71352506729399, −14.04518049091227, −13.56236883670386, −13.02771905513855, −12.46130633725856, −11.92284020410790, −11.38561224389359, −10.73268218379213, −10.24734960825843, −9.613687158403345, −9.018560681642267, −8.671547747032002, −7.705372966118332, −6.892775382919807, −6.495851036386994, −6.121019052184148, −5.301920407509011, −4.640025500565476, −3.914474335335814, −3.212169171676226, −2.431572762088480, −1.317455848207890, −0.7507503571556124,
0.7507503571556124, 1.317455848207890, 2.431572762088480, 3.212169171676226, 3.914474335335814, 4.640025500565476, 5.301920407509011, 6.121019052184148, 6.495851036386994, 6.892775382919807, 7.705372966118332, 8.671547747032002, 9.018560681642267, 9.613687158403345, 10.24734960825843, 10.73268218379213, 11.38561224389359, 11.92284020410790, 12.46130633725856, 13.02771905513855, 13.56236883670386, 14.04518049091227, 14.71352506729399, 15.36425044428285, 15.75780241604222
