| L(s) = 1 | − 3-s − 3·5-s + 9-s + 11-s + 13-s + 3·15-s + 4·17-s + 3·19-s + 2·23-s + 4·25-s − 27-s − 5·29-s − 33-s − 9·37-s − 39-s − 10·43-s − 3·45-s − 5·47-s − 4·51-s + 6·53-s − 3·55-s − 3·57-s + 13·59-s + 6·61-s − 3·65-s + 67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.774·15-s + 0.970·17-s + 0.688·19-s + 0.417·23-s + 4/5·25-s − 0.192·27-s − 0.928·29-s − 0.174·33-s − 1.47·37-s − 0.160·39-s − 1.52·43-s − 0.447·45-s − 0.729·47-s − 0.560·51-s + 0.824·53-s − 0.404·55-s − 0.397·57-s + 1.69·59-s + 0.768·61-s − 0.372·65-s + 0.122·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.96682491647261, −13.37580075871842, −12.93511292300727, −12.30967006175533, −11.88052475754772, −11.66203544135029, −11.16439173690805, −10.63686385933055, −10.09051213263696, −9.600778376271295, −8.978580693560247, −8.369239123691100, −7.961469418027166, −7.472535051375168, −6.830071241512934, −6.659554579161799, −5.655152638941797, −5.287185343115240, −4.833156892983767, −3.979832895239152, −3.588721496047189, −3.285850385183474, −2.242966051257446, −1.407890717212535, −0.7710232063163113, 0,
0.7710232063163113, 1.407890717212535, 2.242966051257446, 3.285850385183474, 3.588721496047189, 3.979832895239152, 4.833156892983767, 5.287185343115240, 5.655152638941797, 6.659554579161799, 6.830071241512934, 7.472535051375168, 7.961469418027166, 8.369239123691100, 8.978580693560247, 9.600778376271295, 10.09051213263696, 10.63686385933055, 11.16439173690805, 11.66203544135029, 11.88052475754772, 12.30967006175533, 12.93511292300727, 13.37580075871842, 13.96682491647261
