| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s − 2·11-s − 4·15-s + 6·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s − 27-s − 6·29-s + 10·31-s + 2·33-s − 8·35-s + 4·37-s − 4·43-s + 4·45-s − 2·47-s − 3·49-s − 6·51-s + 10·53-s − 8·55-s − 2·57-s + 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.03·15-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.348·33-s − 1.35·35-s + 0.657·37-s − 0.609·43-s + 0.596·45-s − 0.291·47-s − 3/7·49-s − 0.840·51-s + 1.37·53-s − 1.07·55-s − 0.264·57-s + 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.467429780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.467429780\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.85780159842082, −14.50967994761857, −13.82667415918889, −13.39918468636846, −13.08578991420629, −12.45263191665843, −11.91118964935638, −11.40264130940519, −10.44287397104153, −10.12351770254739, −9.817840722718835, −9.490243743266433, −8.545651463078140, −7.998693972592432, −7.218673713625652, −6.643743123984608, −5.930087050128397, −5.766311393296891, −5.257080845685625, −4.461836076398720, −3.567296192165818, −2.847111868805994, −2.210541866911405, −1.454273137193293, −0.6280561156806411,
0.6280561156806411, 1.454273137193293, 2.210541866911405, 2.847111868805994, 3.567296192165818, 4.461836076398720, 5.257080845685625, 5.766311393296891, 5.930087050128397, 6.643743123984608, 7.218673713625652, 7.998693972592432, 8.545651463078140, 9.490243743266433, 9.817840722718835, 10.12351770254739, 10.44287397104153, 11.40264130940519, 11.91118964935638, 12.45263191665843, 13.08578991420629, 13.39918468636846, 13.82667415918889, 14.50967994761857, 14.85780159842082
