| L(s) = 1 | + 3-s − 2·4-s + 9-s + 4·11-s − 2·12-s − 2·13-s + 4·16-s + 3·17-s + 7·19-s − 5·23-s + 27-s − 29-s + 4·33-s − 2·36-s + 8·37-s − 2·39-s + 5·41-s + 4·43-s − 8·44-s − 7·47-s + 4·48-s + 3·51-s + 4·52-s + 53-s + 7·57-s − 2·59-s + 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s + 1.20·11-s − 0.577·12-s − 0.554·13-s + 16-s + 0.727·17-s + 1.60·19-s − 1.04·23-s + 0.192·27-s − 0.185·29-s + 0.696·33-s − 1/3·36-s + 1.31·37-s − 0.320·39-s + 0.780·41-s + 0.609·43-s − 1.20·44-s − 1.02·47-s + 0.577·48-s + 0.420·51-s + 0.554·52-s + 0.137·53-s + 0.927·57-s − 0.260·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.059420323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.059420323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−13.72853045859025, −13.34813795419225, −12.75962172071308, −12.20150720625390, −11.91766054516510, −11.37222280550999, −10.64410117580667, −9.966390326650997, −9.546886452219568, −9.446053669435579, −8.871713250887331, −8.071983425929616, −7.909220946039404, −7.333024945001405, −6.707741268563956, −5.956886196688483, −5.566866912881983, −4.909391708159061, −4.276830422017540, −3.918370105120172, −3.288629564001746, −2.794700225984443, −1.857104265871128, −1.177864425867539, −0.6022375111243052,
0.6022375111243052, 1.177864425867539, 1.857104265871128, 2.794700225984443, 3.288629564001746, 3.918370105120172, 4.276830422017540, 4.909391708159061, 5.566866912881983, 5.956886196688483, 6.707741268563956, 7.333024945001405, 7.909220946039404, 8.071983425929616, 8.871713250887331, 9.446053669435579, 9.546886452219568, 9.966390326650997, 10.64410117580667, 11.37222280550999, 11.91766054516510, 12.20150720625390, 12.75962172071308, 13.34813795419225, 13.72853045859025
