| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 11-s + 12-s + 2·13-s + 4·15-s + 16-s − 2·17-s − 18-s − 2·19-s + 4·20-s − 22-s − 6·23-s − 24-s + 11·25-s − 2·26-s + 27-s − 4·30-s − 10·31-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.894·20-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s + 0.192·27-s − 0.730·30-s − 1.79·31-s − 0.176·32-s + 0.174·33-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 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.77711726036158, −14.14678874387861, −13.57906628201990, −13.28010564287885, −12.75751058807057, −12.19031225571916, −11.41991934486779, −10.85059026507654, −10.46763779664252, −9.745377640202940, −9.627379419295678, −9.001561338596606, −8.559609094413820, −8.102769895980264, −7.227262979913388, −6.836491373311561, −6.070448651888014, −5.912761892318917, −5.174813056055327, −4.316610453381424, −3.686140421703705, −2.827916868958681, −2.279705704120218, −1.707317028516404, −1.280989087513553, 0,
1.280989087513553, 1.707317028516404, 2.279705704120218, 2.827916868958681, 3.686140421703705, 4.316610453381424, 5.174813056055327, 5.912761892318917, 6.070448651888014, 6.836491373311561, 7.227262979913388, 8.102769895980264, 8.559609094413820, 9.001561338596606, 9.627379419295678, 9.745377640202940, 10.46763779664252, 10.85059026507654, 11.41991934486779, 12.19031225571916, 12.75751058807057, 13.28010564287885, 13.57906628201990, 14.14678874387861, 14.77711726036158
