| L(s) = 1 | − 3-s − 5·7-s + 9-s − 11-s + 13-s − 3·17-s + 4·19-s + 5·21-s − 6·23-s − 27-s + 7·29-s + 31-s + 33-s − 12·37-s − 39-s − 10·41-s + 8·43-s − 11·47-s + 18·49-s + 3·51-s − 9·53-s − 4·57-s − 3·59-s − 11·61-s − 5·63-s + 9·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 1.09·21-s − 1.25·23-s − 0.192·27-s + 1.29·29-s + 0.179·31-s + 0.174·33-s − 1.97·37-s − 0.160·39-s − 1.56·41-s + 1.21·43-s − 1.60·47-s + 18/7·49-s + 0.420·51-s − 1.23·53-s − 0.529·57-s − 0.390·59-s − 1.40·61-s − 0.629·63-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.93415622179630, −13.47711054281055, −13.02650169610423, −12.50890250663728, −12.07679210462877, −11.82626756112394, −11.04493312348455, −10.46676618468525, −10.21559559386801, −9.688079889606725, −9.278421705631704, −8.712207552288155, −8.096295639982530, −7.532322213911290, −6.811351218939889, −6.512546900122553, −6.218654966111411, −5.507865036059438, −5.022338794244973, −4.370681206578607, −3.642012291633993, −3.277092092142128, −2.703491949505977, −1.896532403442343, −1.113346062021959, 0, 0,
1.113346062021959, 1.896532403442343, 2.703491949505977, 3.277092092142128, 3.642012291633993, 4.370681206578607, 5.022338794244973, 5.507865036059438, 6.218654966111411, 6.512546900122553, 6.811351218939889, 7.532322213911290, 8.096295639982530, 8.712207552288155, 9.278421705631704, 9.688079889606725, 10.21559559386801, 10.46676618468525, 11.04493312348455, 11.82626756112394, 12.07679210462877, 12.50890250663728, 13.02650169610423, 13.47711054281055, 13.93415622179630
