| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 7-s + 8-s + 9-s + 3·10-s − 11-s − 12-s + 4·13-s − 14-s − 3·15-s + 16-s − 17-s + 18-s − 19-s + 3·20-s + 21-s − 22-s + 6·23-s − 24-s + 4·25-s + 4·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.670·20-s + 0.218·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.71985343247879, −13.94346793667545, −13.42311296145415, −13.25010800087794, −12.87480874843749, −12.17364666766488, −11.64521502094045, −11.03385457546806, −10.56704381308843, −10.26776687049862, −9.566382384639041, −9.011146945834751, −8.562122166294743, −7.729430076406980, −6.896056654955568, −6.661426069897055, −6.102100595072983, −5.541136614431643, −5.180366195257611, −4.605982309284551, −3.667811481350092, −3.274995015466419, −2.434912525422125, −1.726743042554050, −1.229944752159315, 0,
1.229944752159315, 1.726743042554050, 2.434912525422125, 3.274995015466419, 3.667811481350092, 4.605982309284551, 5.180366195257611, 5.541136614431643, 6.102100595072983, 6.661426069897055, 6.896056654955568, 7.729430076406980, 8.562122166294743, 9.011146945834751, 9.566382384639041, 10.26776687049862, 10.56704381308843, 11.03385457546806, 11.64521502094045, 12.17364666766488, 12.87480874843749, 13.25010800087794, 13.42311296145415, 13.94346793667545, 14.71985343247879
