| L(s) = 1 | + 3-s + 3·5-s + 3·7-s + 9-s − 2·11-s − 4·13-s + 3·15-s − 6·17-s − 5·19-s + 3·21-s − 23-s + 4·25-s + 27-s − 2·29-s + 4·31-s − 2·33-s + 9·35-s + 8·37-s − 4·39-s − 41-s + 4·43-s + 3·45-s − 3·47-s + 2·49-s − 6·51-s + 2·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.774·15-s − 1.45·17-s − 1.14·19-s + 0.654·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s + 1.52·35-s + 1.31·37-s − 0.640·39-s − 0.156·41-s + 0.609·43-s + 0.447·45-s − 0.437·47-s + 2/7·49-s − 0.840·51-s + 0.274·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−13.21952727642625, −13.15792388055696, −12.68907347729944, −11.97586368276174, −11.41821410382220, −11.01098540158233, −10.41580855347142, −10.13405360801138, −9.538586111583187, −9.177646856314394, −8.591468896473354, −8.201506302992710, −7.732422286963932, −7.112671958347638, −6.642879348357145, −6.048001148922669, −5.585307514348497, −4.928025896375466, −4.510894063228912, −4.216032032776639, −3.165695158183730, −2.446785290412125, −2.182747195192732, −1.879645549875124, −0.9801102455063488, 0,
0.9801102455063488, 1.879645549875124, 2.182747195192732, 2.446785290412125, 3.165695158183730, 4.216032032776639, 4.510894063228912, 4.928025896375466, 5.585307514348497, 6.048001148922669, 6.642879348357145, 7.112671958347638, 7.732422286963932, 8.201506302992710, 8.591468896473354, 9.177646856314394, 9.538586111583187, 10.13405360801138, 10.41580855347142, 11.01098540158233, 11.41821410382220, 11.97586368276174, 12.68907347729944, 13.15792388055696, 13.21952727642625
