| L(s) = 1 | − 3-s + 5-s + 9-s − 4·13-s − 15-s − 6·17-s + 2·19-s + 25-s − 27-s + 4·31-s − 10·37-s + 4·39-s + 4·43-s + 45-s − 12·47-s + 6·51-s + 6·53-s − 2·57-s − 12·59-s − 10·61-s − 4·65-s − 4·67-s + 8·73-s − 75-s + 10·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.458·19-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 1.64·37-s + 0.640·39-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 0.488·67-s + 0.936·73-s − 0.115·75-s + 1.12·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.71027241385681, −12.20951632952412, −11.93059634097254, −11.45499301951277, −10.85789598383117, −10.53837001369442, −10.15525016178267, −9.508872894779845, −9.241250905830893, −8.812094712392685, −8.038289171511284, −7.780664824281610, −6.982378568604819, −6.808465801772468, −6.314704430565182, −5.767132613508282, −5.179888792797015, −4.830930273424642, −4.440765056306750, −3.781369699669351, −3.054922013968053, −2.616854864304177, −1.900892770289929, −1.556657396828192, −0.6026786818801371, 0,
0.6026786818801371, 1.556657396828192, 1.900892770289929, 2.616854864304177, 3.054922013968053, 3.781369699669351, 4.440765056306750, 4.830930273424642, 5.179888792797015, 5.767132613508282, 6.314704430565182, 6.808465801772468, 6.982378568604819, 7.780664824281610, 8.038289171511284, 8.812094712392685, 9.241250905830893, 9.508872894779845, 10.15525016178267, 10.53837001369442, 10.85789598383117, 11.45499301951277, 11.93059634097254, 12.20951632952412, 12.71027241385681
