| L(s) = 1 | − 3-s + 4·5-s + 9-s − 5·11-s − 3·13-s − 4·15-s − 3·17-s − 7·19-s − 9·23-s + 11·25-s − 27-s − 2·31-s + 5·33-s + 37-s + 3·39-s − 6·41-s − 4·43-s + 4·45-s − 10·47-s + 3·51-s + 3·53-s − 20·55-s + 7·57-s − 4·59-s + 2·61-s − 12·65-s − 6·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.50·11-s − 0.832·13-s − 1.03·15-s − 0.727·17-s − 1.60·19-s − 1.87·23-s + 11/5·25-s − 0.192·27-s − 0.359·31-s + 0.870·33-s + 0.164·37-s + 0.480·39-s − 0.937·41-s − 0.609·43-s + 0.596·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s − 2.69·55-s + 0.927·57-s − 0.520·59-s + 0.256·61-s − 1.48·65-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2573045298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2573045298\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.77202352441677, −13.33623765162295, −12.99722065240773, −12.55733717533788, −12.05578258889376, −11.33323811500258, −10.69629616223696, −10.40638270480873, −10.01311880351500, −9.617217935567348, −9.004919366603936, −8.247636800220232, −8.014005202705644, −7.076560734488422, −6.600711936314394, −6.196917445681685, −5.607450245716679, −5.238678130891250, −4.682293300193045, −4.141274538286134, −3.094742845613022, −2.364277573203054, −2.086493741805301, −1.531944276014030, −0.1516330129262388,
0.1516330129262388, 1.531944276014030, 2.086493741805301, 2.364277573203054, 3.094742845613022, 4.141274538286134, 4.682293300193045, 5.238678130891250, 5.607450245716679, 6.196917445681685, 6.600711936314394, 7.076560734488422, 8.014005202705644, 8.247636800220232, 9.004919366603936, 9.617217935567348, 10.01311880351500, 10.40638270480873, 10.69629616223696, 11.33323811500258, 12.05578258889376, 12.55733717533788, 12.99722065240773, 13.33623765162295, 13.77202352441677
