| L(s) = 1 | − 2·4-s + 5-s − 5·11-s + 4·16-s − 3·17-s − 2·19-s − 2·20-s + 4·23-s + 25-s − 5·29-s + 2·31-s + 4·37-s + 2·41-s − 8·43-s + 10·44-s − 9·47-s + 2·53-s − 5·55-s + 8·59-s − 8·61-s − 8·64-s − 10·67-s + 6·68-s − 5·71-s + 9·73-s + 4·76-s − 8·79-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.50·11-s + 16-s − 0.727·17-s − 0.458·19-s − 0.447·20-s + 0.834·23-s + 1/5·25-s − 0.928·29-s + 0.359·31-s + 0.657·37-s + 0.312·41-s − 1.21·43-s + 1.50·44-s − 1.31·47-s + 0.274·53-s − 0.674·55-s + 1.04·59-s − 1.02·61-s − 64-s − 1.22·67-s + 0.727·68-s − 0.593·71-s + 1.05·73-s + 0.458·76-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−13.02734149598874, −12.85212324995100, −12.09430074765409, −11.58203823533483, −10.98737719163811, −10.67145767095903, −10.21475101821275, −9.658180890778800, −9.486426123821313, −8.762767436366546, −8.446971768504124, −8.083044613301261, −7.475782473896933, −7.035663692844487, −6.436409488679941, −5.799601296274241, −5.514494162163404, −4.911259642503696, −4.622245968523649, −4.093312263062874, −3.371631436906199, −2.885384944610004, −2.393384532571207, −1.678890959570879, −1.067108687187865, 0, 0,
1.067108687187865, 1.678890959570879, 2.393384532571207, 2.885384944610004, 3.371631436906199, 4.093312263062874, 4.622245968523649, 4.911259642503696, 5.514494162163404, 5.799601296274241, 6.436409488679941, 7.035663692844487, 7.475782473896933, 8.083044613301261, 8.446971768504124, 8.762767436366546, 9.486426123821313, 9.658180890778800, 10.21475101821275, 10.67145767095903, 10.98737719163811, 11.58203823533483, 12.09430074765409, 12.85212324995100, 13.02734149598874
