| L(s) = 1 | + 3-s − 3·5-s + 7-s + 9-s − 4·11-s − 3·13-s − 3·15-s + 21-s − 23-s + 4·25-s + 27-s − 29-s − 2·31-s − 4·33-s − 3·35-s + 5·37-s − 3·39-s + 5·41-s + 7·43-s − 3·45-s − 3·47-s + 49-s − 12·53-s + 12·55-s + 2·59-s + 6·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s − 0.774·15-s + 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.185·29-s − 0.359·31-s − 0.696·33-s − 0.507·35-s + 0.821·37-s − 0.480·39-s + 0.780·41-s + 1.06·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 1.64·53-s + 1.61·55-s + 0.260·59-s + 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−15.42577608406140, −14.79635418047622, −14.45988430231449, −13.91336027371669, −13.10729753154625, −12.70866595658722, −12.29434298013719, −11.60393320710353, −11.06393281826729, −10.73337950536361, −9.862006807668377, −9.508735172007834, −8.698427388240372, −8.112017683711596, −7.735273882978326, −7.465647983517502, −6.730930778864516, −5.867485021098046, −5.062976103548784, −4.659755033774829, −3.949039769123647, −3.396423477697042, −2.590600925124636, −2.119104845244256, −0.8743018290997513, 0,
0.8743018290997513, 2.119104845244256, 2.590600925124636, 3.396423477697042, 3.949039769123647, 4.659755033774829, 5.062976103548784, 5.867485021098046, 6.730930778864516, 7.465647983517502, 7.735273882978326, 8.112017683711596, 8.698427388240372, 9.508735172007834, 9.862006807668377, 10.73337950536361, 11.06393281826729, 11.60393320710353, 12.29434298013719, 12.70866595658722, 13.10729753154625, 13.91336027371669, 14.45988430231449, 14.79635418047622, 15.42577608406140
