| L(s) = 1 | + 3-s − 7-s + 9-s − 13-s − 2·17-s + 4·19-s − 21-s − 8·23-s + 27-s − 2·29-s − 8·31-s + 2·37-s − 39-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 2·51-s − 10·53-s + 4·57-s + 12·59-s + 10·61-s − 63-s − 12·67-s − 8·69-s − 2·73-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.125·63-s − 1.46·67-s − 0.963·69-s − 0.234·73-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.09307745580607, −12.96816573374672, −12.28908584454670, −11.83102475479270, −11.41281349206597, −10.87707646725518, −10.29809985536084, −9.817625106760155, −9.522806747456429, −9.022523239488784, −8.547826279574385, −7.871566238616944, −7.641008403784658, −7.134158745542599, −6.422893620181817, −6.164190382681290, −5.377072548011588, −5.044059360544386, −4.253459353588303, −3.829558188600159, −3.340732254664665, −2.767034902496803, −2.032012717285471, −1.751048233388960, −0.7527071386864842, 0,
0.7527071386864842, 1.751048233388960, 2.032012717285471, 2.767034902496803, 3.340732254664665, 3.829558188600159, 4.253459353588303, 5.044059360544386, 5.377072548011588, 6.164190382681290, 6.422893620181817, 7.134158745542599, 7.641008403784658, 7.871566238616944, 8.547826279574385, 9.022523239488784, 9.522806747456429, 9.817625106760155, 10.29809985536084, 10.87707646725518, 11.41281349206597, 11.83102475479270, 12.28908584454670, 12.96816573374672, 13.09307745580607
