| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 6·17-s − 4·23-s − 25-s + 27-s + 6·29-s + 4·31-s − 4·33-s + 37-s − 2·39-s − 6·41-s − 8·43-s − 2·45-s − 8·47-s − 7·49-s − 6·51-s + 6·53-s + 8·55-s + 14·61-s + 4·65-s + 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.164·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s + 1.07·55-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 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 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 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.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.838577897119997638086467874831, −8.539318554496821148342136635380, −8.187225933028884364083657365708, −7.30237423640016640017209298342, −6.43740522898978385421421574850, −5.02682421877434292524821371969, −4.29748430655277376425621516910, −3.15107523746645431705635975623, −2.15267051715435814685355305724, 0,
2.15267051715435814685355305724, 3.15107523746645431705635975623, 4.29748430655277376425621516910, 5.02682421877434292524821371969, 6.43740522898978385421421574850, 7.30237423640016640017209298342, 8.187225933028884364083657365708, 8.539318554496821148342136635380, 9.838577897119997638086467874831
