| L(s) = 1 | + 3-s − 2·7-s − 2·9-s + 3·11-s + 2·13-s + 3·17-s + 19-s − 2·21-s + 6·23-s − 5·27-s − 4·31-s + 3·33-s + 2·37-s + 2·39-s − 43-s + 6·47-s − 3·49-s + 3·51-s − 6·53-s + 57-s + 7·61-s + 4·63-s + 8·67-s + 6·69-s + 3·71-s − 8·73-s − 6·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s − 0.436·21-s + 1.25·23-s − 0.962·27-s − 0.718·31-s + 0.522·33-s + 0.328·37-s + 0.320·39-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.420·51-s − 0.824·53-s + 0.132·57-s + 0.896·61-s + 0.503·63-s + 0.977·67-s + 0.722·69-s + 0.356·71-s − 0.936·73-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 229 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.78586489161387, −12.35448059242613, −11.80018159199815, −11.29358796254803, −11.11080373404745, −10.43119817663013, −9.864215619173677, −9.523659983541577, −9.026375597794456, −8.769041608715593, −8.279522453517540, −7.672121742893757, −7.252928214927544, −6.762301116778766, −6.141089976592471, −5.948221466777158, −5.237181559215461, −4.819154981252492, −3.985398752717952, −3.545361619679373, −3.298060237695006, −2.690548834816754, −2.130941474591046, −1.334893981143443, −0.8638039732665690, 0,
0.8638039732665690, 1.334893981143443, 2.130941474591046, 2.690548834816754, 3.298060237695006, 3.545361619679373, 3.985398752717952, 4.819154981252492, 5.237181559215461, 5.948221466777158, 6.141089976592471, 6.762301116778766, 7.252928214927544, 7.672121742893757, 8.279522453517540, 8.769041608715593, 9.026375597794456, 9.523659983541577, 9.864215619173677, 10.43119817663013, 11.11080373404745, 11.29358796254803, 11.80018159199815, 12.35448059242613, 12.78586489161387
