| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 11-s + 2·13-s − 2·15-s − 17-s + 4·19-s − 4·21-s − 25-s + 27-s − 2·29-s − 33-s + 8·35-s + 2·37-s + 2·39-s + 6·41-s − 4·43-s − 2·45-s − 12·47-s + 9·49-s − 51-s + 2·53-s + 2·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s − 0.872·21-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 + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 1.75·47-s + 9/7·49-s − 0.140·51-s + 0.274·53-s + 0.269·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 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
−15.28093009010489, −14.77311930267936, −14.04505530769221, −13.54830313937046, −13.06654338650390, −12.71183626823477, −12.08135649000563, −11.48103834131963, −11.06333387322104, −10.22107099707919, −9.850113948231643, −9.288080246957760, −8.817512670280821, −8.082224310758156, −7.689584591946997, −7.083463947623977, −6.502458498261604, −5.966555634513247, −5.200755804793274, −4.402394804715599, −3.730996253924102, −3.340338416792150, −2.826509304778446, −1.931371492131239, −0.8510313509775513, 0,
0.8510313509775513, 1.931371492131239, 2.826509304778446, 3.340338416792150, 3.730996253924102, 4.402394804715599, 5.200755804793274, 5.966555634513247, 6.502458498261604, 7.083463947623977, 7.689584591946997, 8.082224310758156, 8.817512670280821, 9.288080246957760, 9.850113948231643, 10.22107099707919, 11.06333387322104, 11.48103834131963, 12.08135649000563, 12.71183626823477, 13.06654338650390, 13.54830313937046, 14.04505530769221, 14.77311930267936, 15.28093009010489
