| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 4·11-s + 13-s + 16-s − 6·17-s − 4·19-s − 2·20-s + 4·22-s − 25-s + 26-s + 4·29-s + 4·31-s + 32-s − 6·34-s + 12·37-s − 4·38-s − 2·40-s − 12·41-s − 8·43-s + 4·44-s − 2·47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.196·26-s + 0.742·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1.97·37-s − 0.648·38-s − 0.316·40-s − 1.87·41-s − 1.21·43-s + 0.603·44-s − 0.291·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−16.46781651598042, −16.17019218161156, −15.38494118832982, −15.00552581545592, −14.66059112326318, −13.72137050144237, −13.40404429773577, −12.76963152077020, −11.99353698416328, −11.64471237185491, −11.22914289388920, −10.52321644445945, −9.792458349806082, −9.005959832093325, −8.371350814671052, −7.939638614295802, −6.883244768250648, −6.586020908627553, −6.046284418754968, −4.905490667091248, −4.370465790829672, −3.926839410376962, −3.148606834565281, −2.237965947726516, −1.308432402142374, 0,
1.308432402142374, 2.237965947726516, 3.148606834565281, 3.926839410376962, 4.370465790829672, 4.905490667091248, 6.046284418754968, 6.586020908627553, 6.883244768250648, 7.939638614295802, 8.371350814671052, 9.005959832093325, 9.792458349806082, 10.52321644445945, 11.22914289388920, 11.64471237185491, 11.99353698416328, 12.76963152077020, 13.40404429773577, 13.72137050144237, 14.66059112326318, 15.00552581545592, 15.38494118832982, 16.17019218161156, 16.46781651598042
