| L(s) = 1 | + 3-s − 7-s + 9-s − 6·11-s + 4·13-s − 21-s − 6·23-s − 5·25-s + 27-s − 6·29-s − 4·31-s − 6·33-s − 2·37-s + 4·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 6·53-s − 12·59-s − 10·61-s − 63-s + 14·67-s − 6·69-s − 12·71-s + 2·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.218·21-s − 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.125·63-s + 1.71·67-s − 0.722·69-s − 1.42·71-s + 0.234·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.88707687109007, −13.53954973393739, −13.15433041771607, −12.66673156016257, −12.36423278975024, −11.40327034472521, −11.19274756794418, −10.62818043669746, −9.979446137474898, −9.811184220085273, −9.162049228169139, −8.542717598804602, −8.080378468859035, −7.753538847615617, −7.294335855815335, −6.492112028693657, −6.022857446041293, −5.511028691462241, −5.027167743726839, −4.145751360469330, −3.839548195660849, −3.038725330840295, −2.784757456885139, −1.770942747636677, −1.590492840098131, 0, 0,
1.590492840098131, 1.770942747636677, 2.784757456885139, 3.038725330840295, 3.839548195660849, 4.145751360469330, 5.027167743726839, 5.511028691462241, 6.022857446041293, 6.492112028693657, 7.294335855815335, 7.753538847615617, 8.080378468859035, 8.542717598804602, 9.162049228169139, 9.811184220085273, 9.979446137474898, 10.62818043669746, 11.19274756794418, 11.40327034472521, 12.36423278975024, 12.66673156016257, 13.15433041771607, 13.53954973393739, 13.88707687109007
