| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 21-s − 8·23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 10·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.85·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.96789259658743, −12.68200019187941, −12.06528728406687, −11.59709303669179, −11.23253248439074, −10.72086834893602, −10.24466058438904, −9.735204309546503, −9.489211597191647, −8.842201173077444, −8.462114308836363, −8.122654758901512, −7.514055508601964, −7.319898453593579, −6.522021555363779, −6.193684524538993, −5.809962409625532, −4.969544978457743, −4.303211747743904, −3.986368131211801, −3.437663152133810, −2.818362007241213, −2.267422733963210, −1.770947268578736, −1.144928902709292, 0, 0,
1.144928902709292, 1.770947268578736, 2.267422733963210, 2.818362007241213, 3.437663152133810, 3.986368131211801, 4.303211747743904, 4.969544978457743, 5.809962409625532, 6.193684524538993, 6.522021555363779, 7.319898453593579, 7.514055508601964, 8.122654758901512, 8.462114308836363, 8.842201173077444, 9.489211597191647, 9.735204309546503, 10.24466058438904, 10.72086834893602, 11.23253248439074, 11.59709303669179, 12.06528728406687, 12.68200019187941, 12.96789259658743
