| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 3·11-s + 3·13-s + 15-s + 5·19-s − 21-s − 4·23-s − 4·25-s − 27-s − 5·29-s − 2·31-s + 3·33-s − 35-s − 10·37-s − 3·39-s − 45-s − 3·47-s − 6·49-s − 4·53-s + 3·55-s − 5·57-s − 4·59-s − 8·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.832·13-s + 0.258·15-s + 1.14·19-s − 0.218·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s − 0.359·31-s + 0.522·33-s − 0.169·35-s − 1.64·37-s − 0.480·39-s − 0.149·45-s − 0.437·47-s − 6/7·49-s − 0.549·53-s + 0.404·55-s − 0.662·57-s − 0.520·59-s − 1.02·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.86911103366479, −12.48409714376010, −12.01463559299227, −11.62944144997489, −11.12903223953208, −10.86665396427102, −10.41934790046761, −9.699729802590235, −9.590865205165301, −8.848340191343340, −8.190522951903015, −8.051278868224014, −7.435521590265890, −7.132078745844641, −6.437989342467367, −5.930864135721072, −5.478326132452238, −5.136742080339720, −4.556513908044684, −3.964194035929301, −3.475495691665698, −3.071423671411553, −2.174725257162607, −1.648344741683716, −1.146364987098993, 0, 0,
1.146364987098993, 1.648344741683716, 2.174725257162607, 3.071423671411553, 3.475495691665698, 3.964194035929301, 4.556513908044684, 5.136742080339720, 5.478326132452238, 5.930864135721072, 6.437989342467367, 7.132078745844641, 7.435521590265890, 8.051278868224014, 8.190522951903015, 8.848340191343340, 9.590865205165301, 9.699729802590235, 10.41934790046761, 10.86665396427102, 11.12903223953208, 11.62944144997489, 12.01463559299227, 12.48409714376010, 12.86911103366479
