| L(s) = 1 | − 5-s − 7-s + 3·13-s + 7·17-s + 19-s + 5·23-s + 25-s − 5·29-s + 10·31-s + 35-s − 2·37-s − 2·41-s − 6·43-s − 6·49-s + 9·53-s − 7·59-s + 4·61-s − 3·65-s − 7·67-s − 9·73-s − 10·79-s − 2·83-s − 7·85-s + 10·89-s − 3·91-s − 95-s − 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.832·13-s + 1.69·17-s + 0.229·19-s + 1.04·23-s + 1/5·25-s − 0.928·29-s + 1.79·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s − 0.914·43-s − 6/7·49-s + 1.23·53-s − 0.911·59-s + 0.512·61-s − 0.372·65-s − 0.855·67-s − 1.05·73-s − 1.12·79-s − 0.219·83-s − 0.759·85-s + 1.05·89-s − 0.314·91-s − 0.102·95-s − 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.83097949913828, −14.09575318460476, −13.66593224905355, −13.10247698808168, −12.72278239825893, −11.95368461316990, −11.74663863096031, −11.18913587479970, −10.39982327157268, −10.19409394076261, −9.506826765792987, −8.959027678250959, −8.379999833597137, −7.901075382035816, −7.370961701526649, −6.744884974686784, −6.241251769388028, −5.537050348437431, −5.122537397433733, −4.327358195589202, −3.691342564010166, −3.160832322280699, −2.709897614506148, −1.470492407403884, −1.065984144900195, 0,
1.065984144900195, 1.470492407403884, 2.709897614506148, 3.160832322280699, 3.691342564010166, 4.327358195589202, 5.122537397433733, 5.537050348437431, 6.241251769388028, 6.744884974686784, 7.370961701526649, 7.901075382035816, 8.379999833597137, 8.959027678250959, 9.506826765792987, 10.19409394076261, 10.39982327157268, 11.18913587479970, 11.74663863096031, 11.95368461316990, 12.72278239825893, 13.10247698808168, 13.66593224905355, 14.09575318460476, 14.83097949913828
