| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 5·13-s + 15-s − 7·19-s + 21-s − 23-s + 25-s − 27-s + 8·29-s + 5·31-s + 33-s + 35-s + 7·37-s + 5·39-s − 2·41-s − 8·43-s − 45-s + 2·47-s − 6·49-s − 6·53-s + 55-s + 7·57-s + 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s − 1.60·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.898·31-s + 0.174·33-s + 0.169·35-s + 1.15·37-s + 0.800·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s + 0.291·47-s − 6/7·49-s − 0.824·53-s + 0.134·55-s + 0.927·57-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.56863883272210, −12.28481706942915, −11.86372383525821, −11.41422092230392, −10.81181196734646, −10.53631410556206, −9.933588063043740, −9.751399293085330, −9.158467294638469, −8.434341589705547, −8.138748241406894, −7.753452903627505, −7.077444902707834, −6.530823565607176, −6.472451458413472, −5.794901319700051, −5.038850657391046, −4.785209931603437, −4.359684999823990, −3.801582972892667, −3.058363387828296, −2.602134974990329, −2.114039051429414, −1.314979520817769, −0.5198794689244382, 0,
0.5198794689244382, 1.314979520817769, 2.114039051429414, 2.602134974990329, 3.058363387828296, 3.801582972892667, 4.359684999823990, 4.785209931603437, 5.038850657391046, 5.794901319700051, 6.472451458413472, 6.530823565607176, 7.077444902707834, 7.753452903627505, 8.138748241406894, 8.434341589705547, 9.158467294638469, 9.751399293085330, 9.933588063043740, 10.53631410556206, 10.81181196734646, 11.41422092230392, 11.86372383525821, 12.28481706942915, 12.56863883272210
