| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 4·13-s − 14-s + 16-s − 4·19-s − 4·26-s + 28-s + 6·29-s − 10·31-s − 32-s − 8·37-s + 4·38-s + 3·41-s + 43-s + 9·47-s − 6·49-s + 4·52-s − 12·53-s − 56-s − 6·58-s − 6·59-s + 11·61-s + 10·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s − 1.31·37-s + 0.648·38-s + 0.468·41-s + 0.152·43-s + 1.31·47-s − 6/7·49-s + 0.554·52-s − 1.64·53-s − 0.133·56-s − 0.787·58-s − 0.781·59-s + 1.40·61-s + 1.27·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.61902501696444, −14.23151339656595, −13.79427950411885, −12.95717375837495, −12.69758478130058, −12.08860739125368, −11.40247086661555, −11.03707295393776, −10.57317946453346, −10.17590109440436, −9.352972240223342, −8.912182799365157, −8.549754724856485, −7.922327446457383, −7.495944082223447, −6.689192617923730, −6.390240999901236, −5.684026559539389, −5.107412872224182, −4.350283492147895, −3.700851721800994, −3.135378471357808, −2.215816172521502, −1.696424494368034, −0.9510485463545140, 0,
0.9510485463545140, 1.696424494368034, 2.215816172521502, 3.135378471357808, 3.700851721800994, 4.350283492147895, 5.107412872224182, 5.684026559539389, 6.390240999901236, 6.689192617923730, 7.495944082223447, 7.922327446457383, 8.549754724856485, 8.912182799365157, 9.352972240223342, 10.17590109440436, 10.57317946453346, 11.03707295393776, 11.40247086661555, 12.08860739125368, 12.69758478130058, 12.95717375837495, 13.79427950411885, 14.23151339656595, 14.61902501696444
