| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 3·13-s + 2·14-s + 15-s + 16-s + 7·17-s − 18-s − 2·19-s + 20-s − 2·21-s − 22-s − 24-s + 25-s + 3·26-s + 27-s − 2·28-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 − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.832·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.436·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.192·27-s − 0.377·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 89 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.34916556338814, −14.91016186928132, −14.48914747822002, −13.84686023971594, −13.40474179824147, −12.57528076564739, −12.37044347693422, −11.75091712562275, −11.01535899636331, −10.27115846510698, −9.864608588815006, −9.642344322797120, −9.009688871402266, −8.287837422782072, −7.907435271019324, −7.219245431014616, −6.649366373874621, −6.156658605410430, −5.397085298734031, −4.742602844971250, −3.793941450020398, −3.149452852030161, −2.666669452764769, −1.803700041754500, −1.098396271338029, 0,
1.098396271338029, 1.803700041754500, 2.666669452764769, 3.149452852030161, 3.793941450020398, 4.742602844971250, 5.397085298734031, 6.156658605410430, 6.649366373874621, 7.219245431014616, 7.907435271019324, 8.287837422782072, 9.009688871402266, 9.642344322797120, 9.864608588815006, 10.27115846510698, 11.01535899636331, 11.75091712562275, 12.37044347693422, 12.57528076564739, 13.40474179824147, 13.84686023971594, 14.48914747822002, 14.91016186928132, 15.34916556338814
