| L(s) = 1 | + 3-s − 4·7-s − 2·9-s − 3·11-s − 7·13-s − 5·19-s − 4·21-s + 6·23-s − 5·25-s − 5·27-s − 9·29-s − 10·31-s − 3·33-s − 2·37-s − 7·39-s − 11·43-s − 6·47-s + 9·49-s + 6·53-s − 5·57-s + 61-s + 8·63-s + 4·67-s + 6·69-s + 9·71-s − 11·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s − 0.904·11-s − 1.94·13-s − 1.14·19-s − 0.872·21-s + 1.25·23-s − 25-s − 0.962·27-s − 1.67·29-s − 1.79·31-s − 0.522·33-s − 0.328·37-s − 1.12·39-s − 1.67·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.662·57-s + 0.128·61-s + 1.00·63-s + 0.488·67-s + 0.722·69-s + 1.06·71-s − 1.28·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−12.85059852894958, −12.38400103188868, −11.84171069285965, −11.36034652569847, −10.77063091421496, −10.44651697375776, −9.776174159185521, −9.494318842878279, −9.284848710936540, −8.614251206018753, −8.180969099010889, −7.593073114906681, −7.210012449373506, −6.854312478510712, −6.260692690183960, −5.654295799313477, −5.206963044724951, −4.956497005308532, −3.856969499660673, −3.747035625575801, −3.044643318803815, −2.622134667290180, −2.211498331972988, −1.672265697103245, −0.2484943365787438, 0,
0.2484943365787438, 1.672265697103245, 2.211498331972988, 2.622134667290180, 3.044643318803815, 3.747035625575801, 3.856969499660673, 4.956497005308532, 5.206963044724951, 5.654295799313477, 6.260692690183960, 6.854312478510712, 7.210012449373506, 7.593073114906681, 8.180969099010889, 8.614251206018753, 9.284848710936540, 9.494318842878279, 9.776174159185521, 10.44651697375776, 10.77063091421496, 11.36034652569847, 11.84171069285965, 12.38400103188868, 12.85059852894958
