| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s + 8·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s − 10·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 12·47-s − 7·49-s + 2·51-s − 2·53-s + 4·55-s − 4·59-s + 2·61-s − 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s − 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.75·47-s − 49-s + 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 163 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.48663038851965, −12.95226502947315, −12.64042925597032, −12.08026574625027, −11.62857401649669, −11.12072341476338, −10.82544132454447, −10.20488340507430, −9.688768118882396, −9.137107544948368, −8.972960016315644, −8.336625808128541, −7.420662191726109, −7.155218955213930, −6.762841558559053, −6.088877180520280, −5.707439270662698, −5.120164858381402, −4.611349103866026, −4.080969795902935, −3.451507265516944, −2.823223005756909, −2.061882556783832, −1.507721451122364, −0.8741194415942725, 0,
0.8741194415942725, 1.507721451122364, 2.061882556783832, 2.823223005756909, 3.451507265516944, 4.080969795902935, 4.611349103866026, 5.120164858381402, 5.707439270662698, 6.088877180520280, 6.762841558559053, 7.155218955213930, 7.420662191726109, 8.336625808128541, 8.972960016315644, 9.137107544948368, 9.688768118882396, 10.20488340507430, 10.82544132454447, 11.12072341476338, 11.62857401649669, 12.08026574625027, 12.64042925597032, 12.95226502947315, 13.48663038851965
