| L(s) = 1 | − 3·5-s + 7-s + 3·11-s − 13-s − 6·17-s + 4·19-s − 3·23-s + 4·25-s − 3·29-s − 5·31-s − 3·35-s + 2·37-s − 3·41-s + 43-s − 9·47-s − 6·49-s + 6·53-s − 9·55-s − 3·59-s − 13·61-s + 3·65-s + 7·67-s − 12·71-s − 10·73-s + 3·77-s − 11·79-s − 9·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.625·23-s + 4/5·25-s − 0.557·29-s − 0.898·31-s − 0.507·35-s + 0.328·37-s − 0.468·41-s + 0.152·43-s − 1.31·47-s − 6/7·49-s + 0.824·53-s − 1.21·55-s − 0.390·59-s − 1.66·61-s + 0.372·65-s + 0.855·67-s − 1.42·71-s − 1.17·73-s + 0.341·77-s − 1.23·79-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−9.103199191089479312566176670942, −8.440342643149354363470334721625, −7.56110236514010523446597788878, −7.01090509651629700365382030629, −5.96028486844218902790572750178, −4.72711486021756378251653347205, −4.10568373941110424400504581646, −3.18871842909251669274958257865, −1.69415932861805573668432980190, 0,
1.69415932861805573668432980190, 3.18871842909251669274958257865, 4.10568373941110424400504581646, 4.72711486021756378251653347205, 5.96028486844218902790572750178, 7.01090509651629700365382030629, 7.56110236514010523446597788878, 8.440342643149354363470334721625, 9.103199191089479312566176670942
