| L(s) = 1 | + 5-s + 2·7-s + 13-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 + 6·59-s − 14·61-s + 65-s + 2·67-s + 10·71-s + 6·73-s − 4·79-s + 2·83-s − 2·85-s + 14·89-s + 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.277·13-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 + 0.781·59-s − 1.79·61-s + 0.124·65-s + 0.244·67-s + 1.18·71-s + 0.702·73-s − 0.450·79-s + 0.219·83-s − 0.216·85-s + 1.48·89-s + 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.696421774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696421774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 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.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 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.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.93017862148374, −12.18338408413925, −11.82412741300363, −11.26389956124463, −11.00662452207359, −10.47155954384871, −10.01206314728647, −9.466977936811042, −9.041083415413993, −8.602701046881175, −8.096349924152273, −7.662192859735282, −7.107407448143748, −6.543259200644119, −6.251151697878240, −5.464138963628844, −5.157107185649681, −4.644698513103658, −4.195759905262090, −3.244266771109024, −3.186992218828061, −2.136426144630161, −1.892380184711440, −1.142061795598495, −0.5459545092526761,
0.5459545092526761, 1.142061795598495, 1.892380184711440, 2.136426144630161, 3.186992218828061, 3.244266771109024, 4.195759905262090, 4.644698513103658, 5.157107185649681, 5.464138963628844, 6.251151697878240, 6.543259200644119, 7.107407448143748, 7.662192859735282, 8.096349924152273, 8.602701046881175, 9.041083415413993, 9.466977936811042, 10.01206314728647, 10.47155954384871, 11.00662452207359, 11.26389956124463, 11.82412741300363, 12.18338408413925, 12.93017862148374
