| L(s) = 1 | + 2·3-s + 3·5-s + 4·7-s + 9-s + 6·15-s − 3·17-s − 2·19-s + 8·21-s + 6·23-s + 4·25-s − 4·27-s − 9·29-s + 2·31-s + 12·35-s + 7·37-s + 3·41-s − 4·43-s + 3·45-s − 6·47-s + 9·49-s − 6·51-s + 9·53-s − 4·57-s − 5·61-s + 4·63-s + 2·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s + 1.51·7-s + 1/3·9-s + 1.54·15-s − 0.727·17-s − 0.458·19-s + 1.74·21-s + 1.25·23-s + 4/5·25-s − 0.769·27-s − 1.67·29-s + 0.359·31-s + 2.02·35-s + 1.15·37-s + 0.468·41-s − 0.609·43-s + 0.447·45-s − 0.875·47-s + 9/7·49-s − 0.840·51-s + 1.23·53-s − 0.529·57-s − 0.640·61-s + 0.503·63-s + 0.244·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.04518401149662, −12.65836137561966, −11.72661265247149, −11.45056177058675, −10.96589147043980, −10.63104799345618, −9.953629768282217, −9.504325315175842, −9.149229414624557, −8.744675430501721, −8.305782356207046, −7.918449762066293, −7.342221778397637, −6.920171111864083, −6.272927062240473, −5.673751669273234, −5.370232897087525, −4.788588827195428, −4.237565277157654, −3.807006598458937, −2.863088724009964, −2.624915187470704, −2.049054947011970, −1.591547720728772, −1.191718577276700, 0,
1.191718577276700, 1.591547720728772, 2.049054947011970, 2.624915187470704, 2.863088724009964, 3.807006598458937, 4.237565277157654, 4.788588827195428, 5.370232897087525, 5.673751669273234, 6.272927062240473, 6.920171111864083, 7.342221778397637, 7.918449762066293, 8.305782356207046, 8.744675430501721, 9.149229414624557, 9.504325315175842, 9.953629768282217, 10.63104799345618, 10.96589147043980, 11.45056177058675, 11.72661265247149, 12.65836137561966, 13.04518401149662
