| L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·15-s + 2·17-s − 4·19-s − 25-s − 27-s + 6·29-s + 2·37-s − 6·41-s + 12·43-s − 2·45-s − 4·47-s − 7·49-s − 2·51-s + 6·53-s + 4·57-s − 8·59-s − 2·61-s + 4·67-s − 12·71-s + 14·73-s + 75-s + 81-s + 8·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.328·37-s − 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.04·59-s − 0.256·61-s + 0.488·67-s − 1.42·71-s + 1.63·73-s + 0.115·75-s + 1/9·81-s + 0.878·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.64872091674981426056879660984, −6.65271453087389754688678641765, −6.24481419022097484791190557388, −5.33123613268610301036992330808, −4.61954435744172704431317516175, −4.00277049810163717254171221200, −3.23264978418016443173698754103, −2.20668073069374298369043651343, −1.04399565494348541753349774873, 0,
1.04399565494348541753349774873, 2.20668073069374298369043651343, 3.23264978418016443173698754103, 4.00277049810163717254171221200, 4.61954435744172704431317516175, 5.33123613268610301036992330808, 6.24481419022097484791190557388, 6.65271453087389754688678641765, 7.64872091674981426056879660984
