| L(s) = 1 | − 5-s − 7-s − 13-s + 2·19-s − 6·23-s + 25-s + 6·29-s − 8·31-s + 35-s + 10·37-s + 12·41-s − 4·43-s + 49-s − 12·53-s − 6·59-s − 2·61-s + 65-s − 4·67-s − 6·71-s − 10·73-s + 16·79-s + 12·89-s + 91-s − 2·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.277·13-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 1.64·37-s + 1.87·41-s − 0.609·43-s + 1/7·49-s − 1.64·53-s − 0.781·59-s − 0.256·61-s + 0.124·65-s − 0.488·67-s − 0.712·71-s − 1.17·73-s + 1.80·79-s + 1.27·89-s + 0.104·91-s − 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 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.ak |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−12.92656397411023, −12.64376166485072, −12.02026508938499, −11.78572966781661, −11.23017878875895, −10.66797993368771, −10.39165433376478, −9.712054749812079, −9.266085203204213, −9.090705676030575, −8.200072106848781, −7.805959764293332, −7.612371194424555, −6.914981992812529, −6.346902976951554, −5.991425749491484, −5.472025077845074, −4.727539951839956, −4.387546767233298, −3.839056168796908, −3.218168905735165, −2.777187865877610, −2.128934643084011, −1.439215712718246, −0.6895565655801021, 0,
0.6895565655801021, 1.439215712718246, 2.128934643084011, 2.777187865877610, 3.218168905735165, 3.839056168796908, 4.387546767233298, 4.727539951839956, 5.472025077845074, 5.991425749491484, 6.346902976951554, 6.914981992812529, 7.612371194424555, 7.805959764293332, 8.200072106848781, 9.090705676030575, 9.266085203204213, 9.712054749812079, 10.39165433376478, 10.66797993368771, 11.23017878875895, 11.78572966781661, 12.02026508938499, 12.64376166485072, 12.92656397411023
