| L(s) = 1 | − 2·5-s − 2·7-s − 3·9-s − 2·11-s − 7·17-s + 19-s + 5·23-s − 25-s + 2·29-s − 3·31-s + 4·35-s − 7·37-s − 7·41-s + 9·43-s + 6·45-s − 2·47-s − 3·49-s + 6·53-s + 4·55-s + 3·59-s + 11·61-s + 6·63-s − 3·67-s − 16·71-s − 2·73-s + 4·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s − 1.69·17-s + 0.229·19-s + 1.04·23-s − 1/5·25-s + 0.371·29-s − 0.538·31-s + 0.676·35-s − 1.15·37-s − 1.09·41-s + 1.37·43-s + 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s + 0.539·55-s + 0.390·59-s + 1.40·61-s + 0.755·63-s − 0.366·67-s − 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−14.75027603694700, −14.33813576562307, −13.54060467688752, −13.24471597398023, −12.79752046072590, −12.11138106521358, −11.53351006983690, −11.36177549180633, −10.53736856925914, −10.35486838787172, −9.370854082218023, −8.955165454857992, −8.546187222975199, −7.988654599759549, −7.261565395655090, −6.895521551059433, −6.284989541129331, −5.601430717879709, −5.054585525475256, −4.406212933281584, −3.722475557128545, −3.159637736056161, −2.622192949916514, −1.889302719052556, −0.6033133616752534, 0,
0.6033133616752534, 1.889302719052556, 2.622192949916514, 3.159637736056161, 3.722475557128545, 4.406212933281584, 5.054585525475256, 5.601430717879709, 6.284989541129331, 6.895521551059433, 7.261565395655090, 7.988654599759549, 8.546187222975199, 8.955165454857992, 9.370854082218023, 10.35486838787172, 10.53736856925914, 11.36177549180633, 11.53351006983690, 12.11138106521358, 12.79752046072590, 13.24471597398023, 13.54060467688752, 14.33813576562307, 14.75027603694700
