| L(s) = 1 | − 3-s − 4·7-s + 9-s + 13-s + 2·19-s + 4·21-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 2·37-s − 39-s − 6·41-s + 8·43-s + 12·47-s + 9·49-s − 6·53-s − 2·57-s − 12·59-s − 8·61-s − 4·63-s + 2·67-s − 12·71-s + 14·73-s + 5·75-s + 14·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.458·19-s + 0.872·21-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.264·57-s − 1.56·59-s − 1.02·61-s − 0.503·63-s + 0.244·67-s − 1.42·71-s + 1.63·73-s + 0.577·75-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037497494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037497494\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−12.66018007782223, −12.13010810615337, −11.86342722566014, −11.19918860139382, −10.65447805210019, −10.44651088366711, −9.886423633806712, −9.335613741289211, −9.166733106447104, −8.602864525731214, −7.788472375165806, −7.476259467514330, −7.017965596637515, −6.363941267732295, −6.087034429248481, −5.775681120009533, −5.024596996698877, −4.636727025316123, −3.808948724759753, −3.569791781403211, −3.025522736219631, −2.341649222673286, −1.730656846293379, −0.9138878757815680, −0.3342196507946942,
0.3342196507946942, 0.9138878757815680, 1.730656846293379, 2.341649222673286, 3.025522736219631, 3.569791781403211, 3.808948724759753, 4.636727025316123, 5.024596996698877, 5.775681120009533, 6.087034429248481, 6.363941267732295, 7.017965596637515, 7.476259467514330, 7.788472375165806, 8.602864525731214, 9.166733106447104, 9.335613741289211, 9.886423633806712, 10.44651088366711, 10.65447805210019, 11.19918860139382, 11.86342722566014, 12.13010810615337, 12.66018007782223
