| L(s) = 1 | − 2·5-s + 7-s − 2·13-s − 2·17-s + 8·23-s − 25-s + 2·29-s + 4·31-s − 2·35-s − 2·37-s + 6·41-s + 12·43-s − 8·47-s + 49-s + 10·53-s + 4·59-s − 10·61-s + 4·65-s + 4·67-s − 8·71-s + 10·73-s − 4·79-s + 4·85-s + 6·89-s − 2·91-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.450·79-s + 0.433·85-s + 0.635·89-s − 0.209·91-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.71688875748055, −12.18251997314809, −11.86624800658604, −11.36374906526828, −11.02647604844421, −10.55759312425071, −10.14704182705128, −9.366562011843727, −9.179364774541328, −8.614457052666462, −8.102293273044359, −7.712891006180957, −7.277135782206184, −6.831160757955307, −6.364472499892775, −5.651306751670860, −5.227841804544448, −4.598135826075937, −4.348530822686791, −3.776001868034123, −3.136111688267971, −2.625863299778012, −2.151778261572667, −1.245271752386194, −0.7684480336610716, 0,
0.7684480336610716, 1.245271752386194, 2.151778261572667, 2.625863299778012, 3.136111688267971, 3.776001868034123, 4.348530822686791, 4.598135826075937, 5.227841804544448, 5.651306751670860, 6.364472499892775, 6.831160757955307, 7.277135782206184, 7.712891006180957, 8.102293273044359, 8.614457052666462, 9.179364774541328, 9.366562011843727, 10.14704182705128, 10.55759312425071, 11.02647604844421, 11.36374906526828, 11.86624800658604, 12.18251997314809, 12.71688875748055
