| L(s) = 1 | − 2·5-s + 2·7-s − 3·9-s − 6·11-s + 17-s − 19-s + 3·23-s − 25-s − 6·29-s − 9·31-s − 4·35-s + 37-s − 3·41-s + 3·43-s + 6·45-s − 6·47-s − 3·49-s + 6·53-s + 12·55-s + 5·59-s + 7·61-s − 6·63-s + 3·67-s + 8·71-s + 6·73-s − 12·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 9-s − 1.80·11-s + 0.242·17-s − 0.229·19-s + 0.625·23-s − 1/5·25-s − 1.11·29-s − 1.61·31-s − 0.676·35-s + 0.164·37-s − 0.468·41-s + 0.457·43-s + 0.894·45-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.61·55-s + 0.650·59-s + 0.896·61-s − 0.755·63-s + 0.366·67-s + 0.949·71-s + 0.702·73-s − 1.36·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.89727968905440, −14.32644836389886, −13.76491074081426, −13.04015848325927, −12.83861022610002, −12.16481772960294, −11.45965917695916, −11.13758831773239, −10.93787442426814, −10.15287077275743, −9.600748382860582, −8.777467521760650, −8.416869525381391, −7.913351275500421, −7.500152066906022, −7.057102096902125, −6.044746910064079, −5.499290800493252, −5.101856764825320, −4.577729693560684, −3.528836851840039, −3.409575822961894, −2.337631845467193, −2.008199077480847, −0.6950684133303199, 0,
0.6950684133303199, 2.008199077480847, 2.337631845467193, 3.409575822961894, 3.528836851840039, 4.577729693560684, 5.101856764825320, 5.499290800493252, 6.044746910064079, 7.057102096902125, 7.500152066906022, 7.913351275500421, 8.416869525381391, 8.777467521760650, 9.600748382860582, 10.15287077275743, 10.93787442426814, 11.13758831773239, 11.45965917695916, 12.16481772960294, 12.83861022610002, 13.04015848325927, 13.76491074081426, 14.32644836389886, 14.89727968905440
