| L(s) = 1 | − 2·3-s + 3·5-s + 9-s − 6·15-s − 3·17-s + 2·19-s + 6·23-s + 4·25-s + 4·27-s + 9·29-s + 2·31-s − 7·37-s − 3·41-s + 4·43-s + 3·45-s − 6·47-s + 6·51-s + 9·53-s − 4·57-s − 5·61-s − 2·67-s − 12·69-s + 6·71-s + 73-s − 8·75-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s + 1/3·9-s − 1.54·15-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 0.769·27-s + 1.67·29-s + 0.359·31-s − 1.15·37-s − 0.468·41-s + 0.609·43-s + 0.447·45-s − 0.875·47-s + 0.840·51-s + 1.23·53-s − 0.529·57-s − 0.640·61-s − 0.244·67-s − 1.44·69-s + 0.712·71-s + 0.117·73-s − 0.923·75-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 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.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 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
−13.71418700258189, −13.31435442464921, −12.68968282115113, −12.12636033850394, −11.93630066254193, −11.19573951471614, −10.72143930288949, −10.53473981065267, −9.847681846697457, −9.517191383985625, −8.825538777090633, −8.535793990151118, −7.785557341189690, −6.922399141866181, −6.645545394212183, −6.355906999172859, −5.520937051195102, −5.392923789225940, −4.818137080781502, −4.327422516851255, −3.362545119122971, −2.752968089217882, −2.237043482026894, −1.369333419513959, −0.9367026291153257, 0,
0.9367026291153257, 1.369333419513959, 2.237043482026894, 2.752968089217882, 3.362545119122971, 4.327422516851255, 4.818137080781502, 5.392923789225940, 5.520937051195102, 6.355906999172859, 6.645545394212183, 6.922399141866181, 7.785557341189690, 8.535793990151118, 8.825538777090633, 9.517191383985625, 9.847681846697457, 10.53473981065267, 10.72143930288949, 11.19573951471614, 11.93630066254193, 12.12636033850394, 12.68968282115113, 13.31435442464921, 13.71418700258189
