| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 11-s + 2·15-s − 8·17-s − 6·19-s − 4·21-s + 6·23-s − 25-s − 27-s − 2·29-s − 2·31-s − 33-s − 8·35-s − 4·37-s − 2·41-s − 4·43-s − 2·45-s + 4·47-s + 9·49-s + 8·51-s − 8·53-s − 2·55-s + 6·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s − 1.94·17-s − 1.37·19-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.174·33-s − 1.35·35-s − 0.657·37-s − 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s + 1.12·51-s − 1.09·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.18327911475459, −13.61046681973330, −12.93971369264012, −12.76036391384380, −11.90012313458467, −11.60290295922835, −11.16416944326301, −10.86488241868096, −10.48292004542441, −9.605158325669146, −8.913123955392442, −8.553718016094212, −8.209913345449583, −7.464359740222114, −7.028647328794630, −6.587598852048033, −5.930256775646471, −5.112240800950635, −4.832148822791637, −4.180488748448569, −3.972548156745240, −3.000660277386322, −1.991441499339837, −1.822627938861727, −0.7518932342501759, 0,
0.7518932342501759, 1.822627938861727, 1.991441499339837, 3.000660277386322, 3.972548156745240, 4.180488748448569, 4.832148822791637, 5.112240800950635, 5.930256775646471, 6.587598852048033, 7.028647328794630, 7.464359740222114, 8.209913345449583, 8.553718016094212, 8.913123955392442, 9.605158325669146, 10.48292004542441, 10.86488241868096, 11.16416944326301, 11.60290295922835, 11.90012313458467, 12.76036391384380, 12.93971369264012, 13.61046681973330, 14.18327911475459
