| L(s) = 1 | − 5-s − 4·11-s − 13-s + 6·17-s + 3·19-s − 2·23-s − 4·25-s + 6·29-s − 2·31-s + 8·37-s + 2·41-s + 11·43-s − 6·47-s − 7·49-s + 3·53-s + 4·55-s + 4·59-s − 2·61-s + 65-s + 8·67-s − 9·71-s − 10·73-s + 11·83-s − 6·85-s + 2·89-s − 3·95-s + 13·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 0.688·19-s − 0.417·23-s − 4/5·25-s + 1.11·29-s − 0.359·31-s + 1.31·37-s + 0.312·41-s + 1.67·43-s − 0.875·47-s − 49-s + 0.412·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.124·65-s + 0.977·67-s − 1.06·71-s − 1.17·73-s + 1.20·83-s − 0.650·85-s + 0.211·89-s − 0.307·95-s + 1.31·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.242022988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.242022988\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.97744349051755, −12.42175397249839, −11.99363692208919, −11.64136076844268, −11.08734745584125, −10.60882916134570, −9.997431159010454, −9.856262380638577, −9.291652514823611, −8.595715129796777, −8.059362677139566, −7.651936239107225, −7.556907495191903, −6.820490989391243, −6.052323722039156, −5.741385776639371, −5.257884830879581, −4.604178347726113, −4.256678431533017, −3.379793692994112, −3.143345594108880, −2.467078995841529, −1.881722264978091, −0.9654718805052655, −0.4933357856722260,
0.4933357856722260, 0.9654718805052655, 1.881722264978091, 2.467078995841529, 3.143345594108880, 3.379793692994112, 4.256678431533017, 4.604178347726113, 5.257884830879581, 5.741385776639371, 6.052323722039156, 6.820490989391243, 7.556907495191903, 7.651936239107225, 8.059362677139566, 8.595715129796777, 9.291652514823611, 9.856262380638577, 9.997431159010454, 10.60882916134570, 11.08734745584125, 11.64136076844268, 11.99363692208919, 12.42175397249839, 12.97744349051755
