| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s − 11-s − 2·13-s − 2·15-s + 2·17-s + 4·19-s − 21-s − 25-s + 27-s + 6·29-s − 8·31-s − 33-s + 2·35-s − 10·37-s − 2·39-s − 6·41-s − 12·43-s − 2·45-s − 8·47-s + 49-s + 2·51-s − 10·53-s + 2·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s + 0.338·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 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.o |
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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
−8.681664765108636897649884755955, −8.115312456483454877214876052689, −7.32679510087760890751684850820, −6.76740187915684598894074644751, −5.47152315705951695590165198989, −4.70737919284860254794725435359, −3.54444170118099145151017073294, −3.11263104304771233971976876927, −1.70966634244462921375059708809, 0,
1.70966634244462921375059708809, 3.11263104304771233971976876927, 3.54444170118099145151017073294, 4.70737919284860254794725435359, 5.47152315705951695590165198989, 6.76740187915684598894074644751, 7.32679510087760890751684850820, 8.115312456483454877214876052689, 8.681664765108636897649884755955
