| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 13-s − 2·15-s + 4·17-s + 2·19-s + 4·21-s − 6·23-s − 25-s − 27-s + 6·29-s + 6·31-s + 33-s − 8·35-s + 39-s − 6·41-s − 4·43-s + 2·45-s + 12·47-s + 9·49-s − 4·51-s − 4·53-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s + 0.970·17-s + 0.458·19-s + 0.872·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.174·33-s − 1.35·35-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 9/7·49-s − 0.560·51-s − 0.549·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.53312953252247020318079994175, −6.63727601507792723453452977553, −6.22073523783249495447426124614, −5.65498767157657710923053927605, −4.95167198810472972319935829081, −3.93276981342008927171496392599, −3.10335037553970201851117902271, −2.36040497063820719155607451293, −1.18028628730407589941487011259, 0,
1.18028628730407589941487011259, 2.36040497063820719155607451293, 3.10335037553970201851117902271, 3.93276981342008927171496392599, 4.95167198810472972319935829081, 5.65498767157657710923053927605, 6.22073523783249495447426124614, 6.63727601507792723453452977553, 7.53312953252247020318079994175
