| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 11-s + 13-s + 2·15-s + 2·17-s − 4·19-s + 4·21-s − 4·23-s − 25-s − 27-s + 2·29-s + 33-s + 8·35-s + 2·37-s − 39-s + 10·41-s − 4·43-s − 2·45-s + 4·47-s + 9·49-s − 2·51-s − 2·53-s + 2·55-s + 4·57-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 + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.174·33-s + 1.35·35-s + 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.67833644783175, −15.20333429475629, −14.51019791815537, −13.83739800628587, −13.19370895341594, −12.82740739800871, −12.12897328898469, −12.04751053717090, −11.15494724947211, −10.70393343850341, −10.12571485754595, −9.652807175233691, −9.012588975605763, −8.334998952726606, −7.681493663582129, −7.248011737797975, −6.455538003926664, −6.058929937328750, −5.587860596667614, −4.450787352771622, −4.198063253938310, −3.370465103510754, −2.875891998256867, −1.867693190772186, −0.6814345856670516, 0,
0.6814345856670516, 1.867693190772186, 2.875891998256867, 3.370465103510754, 4.198063253938310, 4.450787352771622, 5.587860596667614, 6.058929937328750, 6.455538003926664, 7.248011737797975, 7.681493663582129, 8.334998952726606, 9.012588975605763, 9.652807175233691, 10.12571485754595, 10.70393343850341, 11.15494724947211, 12.04751053717090, 12.12897328898469, 12.82740739800871, 13.19370895341594, 13.83739800628587, 14.51019791815537, 15.20333429475629, 15.67833644783175
