| L(s) = 1 | + 3-s − 4·5-s + 9-s + 2·11-s − 4·15-s − 6·17-s + 2·19-s + 8·23-s + 11·25-s + 27-s + 6·29-s − 10·31-s + 2·33-s + 4·37-s − 4·43-s − 4·45-s + 2·47-s − 6·51-s − 10·53-s − 8·55-s + 2·57-s + 14·59-s + 2·61-s − 2·67-s + 8·69-s − 6·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s − 1.03·15-s − 1.45·17-s + 0.458·19-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 + 0.657·37-s − 0.609·43-s − 0.596·45-s + 0.291·47-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 − 0.244·67-s + 0.963·69-s − 0.712·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.463184511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463184511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 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.ac |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 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.c |
| 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
−12.57746112407627, −11.79706297527611, −11.46771899044386, −11.34109527390672, −10.66147335516955, −10.38306574868010, −9.512233967408521, −9.136795005737157, −8.750246867707545, −8.449785254506301, −7.864226274883197, −7.427429594381165, −6.964204207387819, −6.767525113796400, −6.116457432100873, −5.194253433473822, −4.892722769117100, −4.265655325064920, −3.989716162208139, −3.448229212864918, −2.945107068365981, −2.518901100788706, −1.615402888073645, −1.050237410723959, −0.3339465613894757,
0.3339465613894757, 1.050237410723959, 1.615402888073645, 2.518901100788706, 2.945107068365981, 3.448229212864918, 3.989716162208139, 4.265655325064920, 4.892722769117100, 5.194253433473822, 6.116457432100873, 6.767525113796400, 6.964204207387819, 7.427429594381165, 7.864226274883197, 8.449785254506301, 8.750246867707545, 9.136795005737157, 9.512233967408521, 10.38306574868010, 10.66147335516955, 11.34109527390672, 11.46771899044386, 11.79706297527611, 12.57746112407627
