| L(s) = 1 | − 3-s − 2·4-s + 7-s + 9-s − 11-s + 2·12-s + 13-s + 4·16-s + 17-s − 4·19-s − 21-s + 3·23-s − 27-s − 2·28-s − 8·29-s − 4·31-s + 33-s − 2·36-s − 3·37-s − 39-s − 9·41-s + 8·43-s + 2·44-s − 10·47-s − 4·48-s − 6·49-s − 51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 0.277·13-s + 16-s + 0.242·17-s − 0.917·19-s − 0.218·21-s + 0.625·23-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s − 0.493·37-s − 0.160·39-s − 1.40·41-s + 1.21·43-s + 0.301·44-s − 1.45·47-s − 0.577·48-s − 6/7·49-s − 0.140·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−9.575586729989118658755846906237, −8.794166518054235419755731192379, −8.014291888106917297641637399691, −7.08186018746415518352789859537, −5.92807737483008818768334898563, −5.19390466680147823831440999469, −4.38807723148753791240775707454, −3.38600359730436640893879944525, −1.62360461122545194203515418964, 0,
1.62360461122545194203515418964, 3.38600359730436640893879944525, 4.38807723148753791240775707454, 5.19390466680147823831440999469, 5.92807737483008818768334898563, 7.08186018746415518352789859537, 8.014291888106917297641637399691, 8.794166518054235419755731192379, 9.575586729989118658755846906237
