| L(s) = 1 | + 7-s − 6·11-s − 2·13-s + 4·19-s + 6·23-s − 6·29-s − 8·31-s − 2·37-s − 12·41-s − 4·43-s − 12·47-s + 49-s − 6·53-s − 10·61-s + 8·67-s + 6·71-s + 10·73-s − 6·77-s + 4·79-s + 12·83-s − 12·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s − 0.554·13-s + 0.917·19-s + 1.25·23-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.87·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.28·61-s + 0.977·67-s + 0.712·71-s + 1.17·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131074891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131074891\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.25298834907440, −14.98974060990906, −14.31809637296088, −13.66321134250261, −13.19050696306472, −12.75385972866214, −12.23609553612162, −11.33834387878580, −11.15039149162434, −10.48185303601025, −9.892836451133607, −9.416550840829398, −8.689641968093641, −8.071073146006815, −7.548800615508020, −7.163309634946928, −6.382717902206781, −5.438870083184775, −5.071082745380093, −4.806828470136640, −3.427717940532607, −3.250255660661502, −2.220828334974168, −1.655838393088589, −0.4034098469597023,
0.4034098469597023, 1.655838393088589, 2.220828334974168, 3.250255660661502, 3.427717940532607, 4.806828470136640, 5.071082745380093, 5.438870083184775, 6.382717902206781, 7.163309634946928, 7.548800615508020, 8.071073146006815, 8.689641968093641, 9.416550840829398, 9.892836451133607, 10.48185303601025, 11.15039149162434, 11.33834387878580, 12.23609553612162, 12.75385972866214, 13.19050696306472, 13.66321134250261, 14.31809637296088, 14.98974060990906, 15.25298834907440
