| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 12-s − 6·13-s + 2·14-s + 16-s + 4·17-s − 18-s − 19-s − 2·21-s − 4·23-s − 24-s + 6·26-s + 27-s − 2·28-s − 6·29-s − 6·31-s − 32-s − 4·34-s + 36-s − 10·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.229·19-s − 0.436·21-s − 0.834·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s − 1.64·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−12.77097477957858, −12.42647947977563, −11.78547458117515, −11.58042358896440, −10.78399171904960, −10.24957162988987, −10.00335784807130, −9.715300896996023, −9.002298552692774, −8.918788813517167, −8.222126744626458, −7.607647252431617, −7.351858133335790, −7.069917217902567, −6.397061771085216, −5.766004470430195, −5.420240609132418, −4.807431823802923, −4.071063616167268, −3.571727416108348, −3.158349937015404, −2.485220724225780, −2.052240961776301, −1.547399553557189, −0.5727584399569201, 0,
0.5727584399569201, 1.547399553557189, 2.052240961776301, 2.485220724225780, 3.158349937015404, 3.571727416108348, 4.071063616167268, 4.807431823802923, 5.420240609132418, 5.766004470430195, 6.397061771085216, 7.069917217902567, 7.351858133335790, 7.607647252431617, 8.222126744626458, 8.918788813517167, 9.002298552692774, 9.715300896996023, 10.00335784807130, 10.24957162988987, 10.78399171904960, 11.58042358896440, 11.78547458117515, 12.42647947977563, 12.77097477957858
