| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s + 6·13-s − 2·14-s + 15-s + 16-s + 2·17-s − 18-s − 6·19-s − 20-s − 2·21-s − 6·22-s − 6·23-s + 24-s + 25-s − 6·26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 1.66·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 157470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 157470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153434816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153434816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| 181 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.18317713551921, −12.59977318392771, −12.24523943438204, −11.61952121202458, −11.34464751667155, −11.11004840452929, −10.40465641204924, −10.12219341200579, −9.320781238513640, −8.855047471569950, −8.605823888281931, −7.994064876397275, −7.546386173117686, −6.975239330692236, −6.306221375602339, −6.003543395878446, −5.762140785969770, −4.571080878240996, −4.226196686833700, −3.894275563593797, −3.202454809010260, −2.214798933079934, −1.645698161443335, −1.098089184746673, −0.5899875272470584,
0.5899875272470584, 1.098089184746673, 1.645698161443335, 2.214798933079934, 3.202454809010260, 3.894275563593797, 4.226196686833700, 4.571080878240996, 5.762140785969770, 6.003543395878446, 6.306221375602339, 6.975239330692236, 7.546386173117686, 7.994064876397275, 8.605823888281931, 8.855047471569950, 9.320781238513640, 10.12219341200579, 10.40465641204924, 11.11004840452929, 11.34464751667155, 11.61952121202458, 12.24523943438204, 12.59977318392771, 13.18317713551921
