| L(s) = 1 | + 3-s − 3·5-s + 9-s − 5·11-s − 13-s − 3·15-s − 3·17-s − 7·19-s − 5·23-s + 4·25-s + 27-s − 3·29-s − 4·31-s − 5·33-s − 9·37-s − 39-s + 6·41-s − 43-s − 3·45-s − 3·51-s − 10·53-s + 15·55-s − 7·57-s + 10·59-s + 5·61-s + 3·65-s − 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.774·15-s − 0.727·17-s − 1.60·19-s − 1.04·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.718·31-s − 0.870·33-s − 1.47·37-s − 0.160·39-s + 0.937·41-s − 0.152·43-s − 0.447·45-s − 0.420·51-s − 1.37·53-s + 2.02·55-s − 0.927·57-s + 1.30·59-s + 0.640·61-s + 0.372·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−12.93566863478934, −12.74301148962513, −12.33051464591492, −11.65438114952033, −11.23861248892434, −10.79742528425814, −10.39375285918001, −9.936790381269238, −9.299718973761690, −8.706678378323699, −8.296710768313319, −8.076001527342817, −7.520129303000401, −7.092858867913974, −6.660616112843421, −5.883003073735709, −5.387943270324186, −4.754237310934170, −4.215596244575982, −3.948969389844057, −3.301544117628302, −2.712946801283000, −2.146021111047338, −1.719997586358880, −0.4353673280406069, 0,
0.4353673280406069, 1.719997586358880, 2.146021111047338, 2.712946801283000, 3.301544117628302, 3.948969389844057, 4.215596244575982, 4.754237310934170, 5.387943270324186, 5.883003073735709, 6.660616112843421, 7.092858867913974, 7.520129303000401, 8.076001527342817, 8.296710768313319, 8.706678378323699, 9.299718973761690, 9.936790381269238, 10.39375285918001, 10.79742528425814, 11.23861248892434, 11.65438114952033, 12.33051464591492, 12.74301148962513, 12.93566863478934
