| L(s) = 1 | + 4·7-s − 3·11-s − 4·13-s − 3·17-s − 5·19-s − 6·23-s − 6·29-s + 2·31-s − 4·37-s − 3·41-s + 11·43-s + 9·49-s + 6·53-s + 3·59-s + 10·61-s + 5·67-s + 6·71-s + 7·73-s − 12·77-s + 14·79-s + 12·83-s + 6·89-s − 16·91-s − 11·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.904·11-s − 1.10·13-s − 0.727·17-s − 1.14·19-s − 1.25·23-s − 1.11·29-s + 0.359·31-s − 0.657·37-s − 0.468·41-s + 1.67·43-s + 9/7·49-s + 0.824·53-s + 0.390·59-s + 1.28·61-s + 0.610·67-s + 0.712·71-s + 0.819·73-s − 1.36·77-s + 1.57·79-s + 1.31·83-s + 0.635·89-s − 1.67·91-s − 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−13.71742654021115, −13.33717530647249, −12.65282470523361, −12.32719739875298, −11.79109467433495, −11.28143408864453, −10.82299894303364, −10.46622899821262, −9.955600453604225, −9.302951903106034, −8.800804890746606, −8.200783640682965, −7.885849675376908, −7.506386113103423, −6.816951896231256, −6.323969207343169, −5.430139452814533, −5.276859342442011, −4.667383388074208, −4.119679124248140, −3.682069334755709, −2.492558117349717, −2.234311664712997, −1.861932324707784, −0.7684303896182987, 0,
0.7684303896182987, 1.861932324707784, 2.234311664712997, 2.492558117349717, 3.682069334755709, 4.119679124248140, 4.667383388074208, 5.276859342442011, 5.430139452814533, 6.323969207343169, 6.816951896231256, 7.506386113103423, 7.885849675376908, 8.200783640682965, 8.800804890746606, 9.302951903106034, 9.955600453604225, 10.46622899821262, 10.82299894303364, 11.28143408864453, 11.79109467433495, 12.32719739875298, 12.65282470523361, 13.33717530647249, 13.71742654021115
