| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s − 13-s + 6·17-s − 5·19-s − 2·21-s − 6·23-s − 27-s − 29-s − 10·31-s + 4·33-s − 3·37-s + 39-s − 7·41-s + 8·43-s − 7·47-s − 3·49-s − 6·51-s + 3·53-s + 5·57-s + 6·59-s + 2·61-s + 2·63-s + 9·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.45·17-s − 1.14·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s − 0.185·29-s − 1.79·31-s + 0.696·33-s − 0.493·37-s + 0.160·39-s − 1.09·41-s + 1.21·43-s − 1.02·47-s − 3/7·49-s − 0.840·51-s + 0.412·53-s + 0.662·57-s + 0.781·59-s + 0.256·61-s + 0.251·63-s + 1.09·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7349709656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7349709656\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.39792200686339, −12.96785836756517, −12.53088324376397, −12.11112279750284, −11.60248076906741, −11.07013244465218, −10.62964947682819, −10.22083528210120, −9.832331471218505, −9.161993277891606, −8.492430037252884, −7.970906267160312, −7.734380946980046, −7.147425714077872, −6.511397848299733, −5.879592035333954, −5.370117642752237, −5.126619376103835, −4.474393532820302, −3.758087410920475, −3.352989526380486, −2.288246994775232, −2.023735052431686, −1.232978584911185, −0.2730572940198143,
0.2730572940198143, 1.232978584911185, 2.023735052431686, 2.288246994775232, 3.352989526380486, 3.758087410920475, 4.474393532820302, 5.126619376103835, 5.370117642752237, 5.879592035333954, 6.511397848299733, 7.147425714077872, 7.734380946980046, 7.970906267160312, 8.492430037252884, 9.161993277891606, 9.832331471218505, 10.22083528210120, 10.62964947682819, 11.07013244465218, 11.60248076906741, 12.11112279750284, 12.53088324376397, 12.96785836756517, 13.39792200686339
