| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s − 15-s + 2·17-s − 4·19-s − 21-s + 25-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 35-s − 6·37-s − 39-s + 10·41-s − 4·43-s − 45-s + 49-s + 2·51-s − 6·53-s − 4·55-s − 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717310267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717310267\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.14113255917567, −13.29403444647512, −12.92935907446561, −12.44332391247263, −12.03562914033444, −11.45349921194862, −10.84307412382718, −10.56146817227062, −9.586441823024621, −9.492983790018381, −8.916454911525419, −8.421570205461774, −7.831822911803556, −7.238850415451973, −6.945112810766869, −6.231907200647270, −5.736207429100915, −5.036009145666134, −4.210720156064118, −3.966883058172684, −3.341095450188736, −2.798390330033828, −1.873477822245151, −1.484269631991217, −0.3951138081343264,
0.3951138081343264, 1.484269631991217, 1.873477822245151, 2.798390330033828, 3.341095450188736, 3.966883058172684, 4.210720156064118, 5.036009145666134, 5.736207429100915, 6.231907200647270, 6.945112810766869, 7.238850415451973, 7.831822911803556, 8.421570205461774, 8.916454911525419, 9.492983790018381, 9.586441823024621, 10.56146817227062, 10.84307412382718, 11.45349921194862, 12.03562914033444, 12.44332391247263, 12.92935907446561, 13.29403444647512, 14.14113255917567
