| L(s) = 1 | − 2-s − 4-s + 3·8-s − 16-s + 2·17-s − 4·19-s − 6·23-s − 5·25-s + 4·29-s − 5·32-s − 2·34-s + 10·37-s + 4·38-s − 12·41-s − 4·43-s + 6·46-s + 10·47-s + 5·50-s + 12·53-s − 4·58-s + 14·59-s + 10·61-s + 7·64-s − 2·68-s + 8·71-s + 2·73-s − 10·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 1.25·23-s − 25-s + 0.742·29-s − 0.883·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s − 1.87·41-s − 0.609·43-s + 0.884·46-s + 1.45·47-s + 0.707·50-s + 1.64·53-s − 0.525·58-s + 1.82·59-s + 1.28·61-s + 7/8·64-s − 0.242·68-s + 0.949·71-s + 0.234·73-s − 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047550205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047550205\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.90073178601218, −13.63368332163635, −13.19081374915504, −12.54681929086003, −12.04792933268320, −11.55819883797270, −10.98683149448379, −10.18085358318591, −10.11806290203971, −9.658893966615106, −8.909095931188454, −8.440284896623465, −8.113059198344208, −7.606276422784615, −6.904480108688056, −6.418737799572568, −5.617230378937600, −5.292918875495633, −4.420766748327747, −4.015237770216656, −3.538962480478400, −2.419716331989901, −2.030326816057131, −1.094918024577278, −0.4385553328531176,
0.4385553328531176, 1.094918024577278, 2.030326816057131, 2.419716331989901, 3.538962480478400, 4.015237770216656, 4.420766748327747, 5.292918875495633, 5.617230378937600, 6.418737799572568, 6.904480108688056, 7.606276422784615, 8.113059198344208, 8.440284896623465, 8.909095931188454, 9.658893966615106, 10.11806290203971, 10.18085358318591, 10.98683149448379, 11.55819883797270, 12.04792933268320, 12.54681929086003, 13.19081374915504, 13.63368332163635, 13.90073178601218
