| L(s) = 1 | − 2·4-s + 4·7-s + 7·13-s + 4·16-s − 19-s − 8·28-s + 11·31-s + 10·37-s − 5·43-s + 9·49-s − 14·52-s − 61-s − 8·64-s − 5·67-s + 7·73-s + 2·76-s − 13·79-s + 28·91-s − 5·97-s + 13·103-s − 19·109-s + 16·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s + 1.94·13-s + 16-s − 0.229·19-s − 1.51·28-s + 1.97·31-s + 1.64·37-s − 0.762·43-s + 9/7·49-s − 1.94·52-s − 0.128·61-s − 64-s − 0.610·67-s + 0.819·73-s + 0.229·76-s − 1.46·79-s + 2.93·91-s − 0.507·97-s + 1.28·103-s − 1.81·109-s + 1.51·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278941431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278941431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.226489793377276187672579497459, −7.70968740559530548316880656605, −6.49168545191944911536084970286, −5.88224576867942027781627247165, −5.07791215733775506462682583741, −4.40910247858326413136262302082, −3.91363876353059511250004982322, −2.85328542129200108880060222559, −1.54527610827579922329092796392, −0.906817029579863364651722179654,
0.906817029579863364651722179654, 1.54527610827579922329092796392, 2.85328542129200108880060222559, 3.91363876353059511250004982322, 4.40910247858326413136262302082, 5.07791215733775506462682583741, 5.88224576867942027781627247165, 6.49168545191944911536084970286, 7.70968740559530548316880656605, 8.226489793377276187672579497459
