| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 2·11-s − 4·13-s − 4·15-s + 6·17-s − 4·19-s − 25-s − 4·27-s + 4·29-s − 31-s − 4·33-s − 4·37-s − 8·39-s − 10·41-s + 2·43-s − 2·45-s − 8·47-s − 7·49-s + 12·51-s − 4·53-s + 4·55-s − 8·57-s + 8·65-s − 12·67-s + 2·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.03·15-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.769·27-s + 0.742·29-s − 0.179·31-s − 0.696·33-s − 0.657·37-s − 1.28·39-s − 1.56·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 49-s + 1.68·51-s − 0.549·53-s + 0.539·55-s − 1.05·57-s + 0.992·65-s − 1.46·67-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.538758733922093101999460970330, −8.000114529430581410226413786652, −7.58601033116152401988269635391, −6.63079443859875581548723313991, −5.40277194504855072510428472476, −4.57342872063779904798041984921, −3.52159346119794100898696954793, −2.96074059646736536983974611040, −1.88272435199862659869905037397, 0,
1.88272435199862659869905037397, 2.96074059646736536983974611040, 3.52159346119794100898696954793, 4.57342872063779904798041984921, 5.40277194504855072510428472476, 6.63079443859875581548723313991, 7.58601033116152401988269635391, 8.000114529430581410226413786652, 8.538758733922093101999460970330
