| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s − 2·17-s − 2·19-s − 6·23-s − 27-s + 8·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 2·41-s − 4·43-s + 6·47-s − 7·49-s + 2·51-s + 10·53-s + 2·57-s − 8·59-s − 2·61-s + 4·67-s + 6·69-s − 4·73-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.458·19-s − 1.25·23-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.875·47-s − 49-s + 0.280·51-s + 1.37·53-s + 0.264·57-s − 1.04·59-s − 0.256·61-s + 0.488·67-s + 0.722·69-s − 0.468·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517162301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517162301\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−7.63543380932353078797153385892, −6.73032385247620062548122948486, −6.46486637903466601868680275237, −5.76076549179735468117673205524, −4.77469668240256923489336929167, −4.36870222187794946068276381750, −3.55961415157420892591109127168, −2.52031949042821064411796836981, −1.65064319119518825732573742879, −0.61661312961353731560457426542,
0.61661312961353731560457426542, 1.65064319119518825732573742879, 2.52031949042821064411796836981, 3.55961415157420892591109127168, 4.36870222187794946068276381750, 4.77469668240256923489336929167, 5.76076549179735468117673205524, 6.46486637903466601868680275237, 6.73032385247620062548122948486, 7.63543380932353078797153385892
