| L(s) = 1 | − 3·3-s + 6·9-s − 5·11-s + 5·13-s − 7·17-s + 2·19-s − 2·23-s − 9·27-s − 7·29-s + 4·31-s + 15·33-s − 6·37-s − 15·39-s + 12·41-s + 2·43-s − 47-s + 21·51-s − 6·57-s + 4·59-s + 4·61-s − 8·67-s + 6·69-s + 6·73-s + 3·79-s + 9·81-s − 4·83-s + 21·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.50·11-s + 1.38·13-s − 1.69·17-s + 0.458·19-s − 0.417·23-s − 1.73·27-s − 1.29·29-s + 0.718·31-s + 2.61·33-s − 0.986·37-s − 2.40·39-s + 1.87·41-s + 0.304·43-s − 0.145·47-s + 2.94·51-s − 0.794·57-s + 0.520·59-s + 0.512·61-s − 0.977·67-s + 0.722·69-s + 0.702·73-s + 0.337·79-s + 81-s − 0.439·83-s + 2.25·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−14.11526918674892, −13.46969458668573, −13.16159802156580, −12.87786563772831, −12.15512065015880, −11.73669568225328, −11.07090190480642, −10.85806819455907, −10.63599056235986, −9.905457760434397, −9.324968830771289, −8.728480162480104, −8.036636754234313, −7.587193030369635, −6.852909936458900, −6.506324222044055, −5.872799614327187, −5.475512067141962, −5.082050601440265, −4.219666384449032, −4.045991434723704, −2.986942252531515, −2.236415470277040, −1.501630526306967, −0.6570902229918673, 0,
0.6570902229918673, 1.501630526306967, 2.236415470277040, 2.986942252531515, 4.045991434723704, 4.219666384449032, 5.082050601440265, 5.475512067141962, 5.872799614327187, 6.506324222044055, 6.852909936458900, 7.587193030369635, 8.036636754234313, 8.728480162480104, 9.324968830771289, 9.905457760434397, 10.63599056235986, 10.85806819455907, 11.07090190480642, 11.73669568225328, 12.15512065015880, 12.87786563772831, 13.16159802156580, 13.46969458668573, 14.11526918674892
