| L(s) = 1 | − 5-s + 2·7-s − 6·11-s + 2·17-s + 2·19-s + 4·23-s + 25-s − 2·29-s + 2·31-s − 2·35-s − 2·37-s − 6·41-s + 6·47-s − 3·49-s + 2·53-s + 6·55-s − 6·59-s + 14·61-s − 2·67-s − 10·71-s + 6·73-s − 12·77-s + 4·79-s + 2·83-s − 2·85-s − 14·89-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.80·11-s + 0.485·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.875·47-s − 3/7·49-s + 0.274·53-s + 0.809·55-s − 0.781·59-s + 1.79·61-s − 0.244·67-s − 1.18·71-s + 0.702·73-s − 1.36·77-s + 0.450·79-s + 0.219·83-s − 0.216·85-s − 1.48·89-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−15.42725453591916, −14.94693521373504, −14.38076937036295, −13.73599966046756, −13.29815327816420, −12.75152344212221, −12.18258429763588, −11.65466494027542, −11.01682654716032, −10.69542041700231, −10.07718092653546, −9.529058401121502, −8.686510941775531, −8.225856070710477, −7.789851754059183, −7.261430340979986, −6.704296294520679, −5.614376627525746, −5.343871793713748, −4.784101843191849, −4.083258848925221, −3.204280796615018, −2.732354447857418, −1.886083402214610, −0.9825207057741074, 0,
0.9825207057741074, 1.886083402214610, 2.732354447857418, 3.204280796615018, 4.083258848925221, 4.784101843191849, 5.343871793713748, 5.614376627525746, 6.704296294520679, 7.261430340979986, 7.789851754059183, 8.225856070710477, 8.686510941775531, 9.529058401121502, 10.07718092653546, 10.69542041700231, 11.01682654716032, 11.65466494027542, 12.18258429763588, 12.75152344212221, 13.29815327816420, 13.73599966046756, 14.38076937036295, 14.94693521373504, 15.42725453591916
