| L(s) = 1 | + 2·5-s − 4·7-s − 4·11-s − 2·13-s − 6·17-s − 4·19-s − 25-s − 2·29-s − 4·31-s − 8·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s − 8·55-s − 4·59-s − 6·61-s − 4·65-s + 4·67-s − 16·71-s − 6·73-s + 16·77-s + 4·79-s − 12·83-s − 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 1.07·55-s − 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s + 1.82·77-s + 0.450·79-s − 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.41411735591800, −13.10324155189869, −12.86133098416960, −12.25408485414192, −11.77958074088144, −10.91323428102218, −10.64623897871971, −10.23072535779099, −9.735754479191273, −9.257822669244909, −8.912950823498266, −8.375751740367911, −7.549286835716256, −7.197105722890261, −6.665085818777708, −6.098677671875233, −5.766226577981183, −5.287725610584899, −4.428618991883773, −4.150223613438212, −3.238729443084671, −2.757825314127274, −2.222739543589878, −1.852210258547704, −0.5637361579066352, 0,
0.5637361579066352, 1.852210258547704, 2.222739543589878, 2.757825314127274, 3.238729443084671, 4.150223613438212, 4.428618991883773, 5.287725610584899, 5.766226577981183, 6.098677671875233, 6.665085818777708, 7.197105722890261, 7.549286835716256, 8.375751740367911, 8.912950823498266, 9.257822669244909, 9.735754479191273, 10.23072535779099, 10.64623897871971, 10.91323428102218, 11.77958074088144, 12.25408485414192, 12.86133098416960, 13.10324155189869, 13.41411735591800
