| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s + 11-s − 12-s − 2·13-s − 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s − 22-s − 8·23-s + 24-s − 25-s + 2·26-s − 27-s − 29-s + 2·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.185·29-s + 0.365·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.13724049886665, −13.48117142902034, −13.18519880278815, −12.34546218773812, −11.99003130384085, −11.74482608083537, −10.99563751773213, −10.48247884686583, −10.04910235951542, −9.690655711670095, −9.217411924651941, −8.748395478248281, −7.913299928553198, −7.485286375287344, −7.165859969028995, −6.292243217766886, −5.871791960441413, −5.663867227453099, −4.887347029063521, −4.207428814390408, −3.556052192550672, −2.775928439121731, −2.083914751077740, −1.607503644117453, −0.8550278353282138, 0,
0.8550278353282138, 1.607503644117453, 2.083914751077740, 2.775928439121731, 3.556052192550672, 4.207428814390408, 4.887347029063521, 5.663867227453099, 5.871791960441413, 6.292243217766886, 7.165859969028995, 7.485286375287344, 7.913299928553198, 8.748395478248281, 9.217411924651941, 9.690655711670095, 10.04910235951542, 10.48247884686583, 10.99563751773213, 11.74482608083537, 11.99003130384085, 12.34546218773812, 13.18519880278815, 13.48117142902034, 14.13724049886665
