| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 4·11-s − 12-s − 13-s − 16-s + 18-s − 4·19-s − 4·22-s + 8·23-s − 3·24-s − 26-s + 27-s + 2·29-s + 8·31-s + 5·32-s − 4·33-s − 36-s + 6·37-s − 4·38-s − 39-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.852·22-s + 1.66·23-s − 0.612·24-s − 0.196·26-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.696·33-s − 1/6·36-s + 0.986·37-s − 0.648·38-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.490713503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.490713503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 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 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.89360071454049, −12.46344419335436, −12.10212542789400, −11.29146505881785, −11.00991349329518, −10.36865088979074, −10.01506050164708, −9.432246232522099, −9.047632789804008, −8.526908740937905, −8.216549389059472, −7.507142818245635, −7.328747012851629, −6.369720205329296, −6.158513169540508, −5.547735396306720, −4.829504134111865, −4.623482308062136, −4.282701405772051, −3.378767173529794, −3.011624811001526, −2.605356636054970, −2.048651130794546, −1.003514412907971, −0.4925400396295275,
0.4925400396295275, 1.003514412907971, 2.048651130794546, 2.605356636054970, 3.011624811001526, 3.378767173529794, 4.282701405772051, 4.623482308062136, 4.829504134111865, 5.547735396306720, 6.158513169540508, 6.369720205329296, 7.328747012851629, 7.507142818245635, 8.216549389059472, 8.526908740937905, 9.047632789804008, 9.432246232522099, 10.01506050164708, 10.36865088979074, 11.00991349329518, 11.29146505881785, 12.10212542789400, 12.46344419335436, 12.89360071454049
