| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·15-s + 6·17-s + 4·19-s − 25-s + 27-s + 6·29-s + 4·33-s − 6·37-s − 6·41-s + 4·43-s − 2·45-s + 6·51-s − 2·53-s − 8·55-s + 4·57-s + 4·59-s + 2·61-s − 4·67-s − 6·73-s − 75-s + 8·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 0.840·51-s − 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.702·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.54729083195038, −12.67371612942296, −12.21175942106882, −11.94974233101363, −11.65413538057041, −11.01953513560247, −10.33943524659573, −10.05290085968151, −9.412702599185973, −9.097783202423620, −8.496901547693685, −7.996347140763582, −7.716380249210992, −7.115226966077777, −6.701937732058629, −6.137649649031677, −5.413065596189557, −4.996857577333822, −4.276987213737521, −3.777778587064587, −3.434138384185879, −2.952856348512150, −2.175390629517138, −1.283563457724725, −1.057167226229245, 0,
1.057167226229245, 1.283563457724725, 2.175390629517138, 2.952856348512150, 3.434138384185879, 3.777778587064587, 4.276987213737521, 4.996857577333822, 5.413065596189557, 6.137649649031677, 6.701937732058629, 7.115226966077777, 7.716380249210992, 7.996347140763582, 8.496901547693685, 9.097783202423620, 9.412702599185973, 10.05290085968151, 10.33943524659573, 11.01953513560247, 11.65413538057041, 11.94974233101363, 12.21175942106882, 12.67371612942296, 13.54729083195038
