| L(s) = 1 | − 2·7-s + 3·11-s + 2·13-s + 3·17-s + 19-s + 6·23-s − 6·29-s − 4·31-s − 4·37-s + 9·41-s − 43-s + 6·47-s − 3·49-s + 12·53-s − 3·59-s − 8·61-s + 5·67-s − 12·71-s − 11·73-s − 6·77-s − 4·79-s + 12·83-s + 6·89-s − 4·91-s − 5·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s + 1.25·23-s − 1.11·29-s − 0.718·31-s − 0.657·37-s + 1.40·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.390·59-s − 1.02·61-s + 0.610·67-s − 1.42·71-s − 1.28·73-s − 0.683·77-s − 0.450·79-s + 1.31·83-s + 0.635·89-s − 0.419·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.62197788859106, −13.21574683744726, −12.85450964342308, −12.23968524089009, −11.89206364242856, −11.33037895657689, −10.75731268141789, −10.49137568257441, −9.682431597818431, −9.347321813211580, −8.966151854999804, −8.504377209497512, −7.710123938594038, −7.183879276255947, −6.975657618548413, −6.078711508987615, −5.904002112022435, −5.282025918342470, −4.577023097627178, −3.908707694669343, −3.505640897444151, −3.013825964405461, −2.268350071594749, −1.419108993707459, −0.9584672219568971, 0,
0.9584672219568971, 1.419108993707459, 2.268350071594749, 3.013825964405461, 3.505640897444151, 3.908707694669343, 4.577023097627178, 5.282025918342470, 5.904002112022435, 6.078711508987615, 6.975657618548413, 7.183879276255947, 7.710123938594038, 8.504377209497512, 8.966151854999804, 9.347321813211580, 9.682431597818431, 10.49137568257441, 10.75731268141789, 11.33037895657689, 11.89206364242856, 12.23968524089009, 12.85450964342308, 13.21574683744726, 13.62197788859106
