| L(s) = 1 | − 3-s − 2·7-s + 9-s − 2·11-s + 17-s + 19-s + 2·21-s − 23-s − 27-s + 8·31-s + 2·33-s − 37-s + 41-s − 4·43-s − 3·49-s − 51-s + 11·53-s − 57-s + 59-s + 13·61-s − 2·63-s − 12·67-s + 69-s + 13·71-s − 2·73-s + 4·77-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.242·17-s + 0.229·19-s + 0.436·21-s − 0.208·23-s − 0.192·27-s + 1.43·31-s + 0.348·33-s − 0.164·37-s + 0.156·41-s − 0.609·43-s − 3/7·49-s − 0.140·51-s + 1.51·53-s − 0.132·57-s + 0.130·59-s + 1.66·61-s − 0.251·63-s − 1.46·67-s + 0.120·69-s + 1.54·71-s − 0.234·73-s + 0.455·77-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−12.61771366856427, −12.20526874471937, −11.80547504903859, −11.42407194444727, −10.85444051805280, −10.31947457569943, −10.07745535194135, −9.680088513793497, −9.149199606884026, −8.521092617835495, −8.171621672389607, −7.616198902546377, −7.065002124942439, −6.739730315325141, −6.090357136284289, −5.893211482286185, −5.140296802328732, −4.906197314431541, −4.230582400664252, −3.646587230876485, −3.218781471222908, −2.541065026057823, −2.124097264601846, −1.217956278352096, −0.6932311021105538, 0,
0.6932311021105538, 1.217956278352096, 2.124097264601846, 2.541065026057823, 3.218781471222908, 3.646587230876485, 4.230582400664252, 4.906197314431541, 5.140296802328732, 5.893211482286185, 6.090357136284289, 6.739730315325141, 7.065002124942439, 7.616198902546377, 8.171621672389607, 8.521092617835495, 9.149199606884026, 9.680088513793497, 10.07745535194135, 10.31947457569943, 10.85444051805280, 11.42407194444727, 11.80547504903859, 12.20526874471937, 12.61771366856427
