| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 4·13-s + 14-s + 16-s − 6·19-s − 2·20-s − 25-s + 4·26-s + 28-s + 6·29-s − 10·31-s + 32-s − 2·35-s + 6·37-s − 6·38-s − 2·40-s + 2·41-s − 12·43-s − 10·47-s + 49-s − 50-s + 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s − 0.447·20-s − 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 0.338·35-s + 0.986·37-s − 0.973·38-s − 0.316·40-s + 0.312·41-s − 1.82·43-s − 1.45·47-s + 1/7·49-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.40735749541040, −14.10673551014917, −13.30506180180466, −12.83529365031922, −12.69815262462821, −11.85062256538956, −11.34265480797897, −11.18593936080066, −10.65438975744201, −9.940151849912173, −9.405279537242272, −8.481521854966630, −8.237511901398567, −7.935275154962349, −6.964187617754237, −6.652625773212368, −6.139980370912532, −5.322215884473841, −4.946486376255411, −4.158768076652396, −3.785626324632666, −3.371585297971012, −2.406868754579386, −1.844108859896532, −0.9895478820340767, 0,
0.9895478820340767, 1.844108859896532, 2.406868754579386, 3.371585297971012, 3.785626324632666, 4.158768076652396, 4.946486376255411, 5.322215884473841, 6.139980370912532, 6.652625773212368, 6.964187617754237, 7.935275154962349, 8.237511901398567, 8.481521854966630, 9.405279537242272, 9.940151849912173, 10.65438975744201, 11.18593936080066, 11.34265480797897, 11.85062256538956, 12.69815262462821, 12.83529365031922, 13.30506180180466, 14.10673551014917, 14.40735749541040
