| L(s) = 1 | + 2·7-s − 5·11-s − 5·13-s − 17-s + 2·19-s + 23-s + 5·29-s + 31-s + 2·37-s − 6·41-s + 5·43-s + 7·47-s − 3·49-s + 4·53-s + 12·59-s − 6·61-s − 12·67-s − 2·71-s − 12·73-s − 10·77-s − 5·79-s − 2·83-s − 6·89-s − 10·91-s − 16·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.50·11-s − 1.38·13-s − 0.242·17-s + 0.458·19-s + 0.208·23-s + 0.928·29-s + 0.179·31-s + 0.328·37-s − 0.937·41-s + 0.762·43-s + 1.02·47-s − 3/7·49-s + 0.549·53-s + 1.56·59-s − 0.768·61-s − 1.46·67-s − 0.237·71-s − 1.40·73-s − 1.13·77-s − 0.562·79-s − 0.219·83-s − 0.635·89-s − 1.04·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.33774333029815, −13.06139499751620, −12.43012399280473, −12.02763955195442, −11.61014422075987, −11.07651129887557, −10.43455810691510, −10.28316836303289, −9.720836686394370, −9.163247381755744, −8.479204242259917, −8.224232379674034, −7.556939162859706, −7.291405579576698, −6.825767787501677, −5.942543729014979, −5.493920940587879, −5.076210435512130, −4.524320288671999, −4.229955778038804, −3.112338606434310, −2.796380045721471, −2.264627271793694, −1.602494739021511, −0.7431056482285218, 0,
0.7431056482285218, 1.602494739021511, 2.264627271793694, 2.796380045721471, 3.112338606434310, 4.229955778038804, 4.524320288671999, 5.076210435512130, 5.493920940587879, 5.942543729014979, 6.825767787501677, 7.291405579576698, 7.556939162859706, 8.224232379674034, 8.479204242259917, 9.163247381755744, 9.720836686394370, 10.28316836303289, 10.43455810691510, 11.07651129887557, 11.61014422075987, 12.02763955195442, 12.43012399280473, 13.06139499751620, 13.33774333029815
