| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 11-s + 6·13-s + 16-s + 6·17-s − 20-s − 22-s + 4·23-s + 25-s + 6·26-s − 2·29-s + 32-s + 6·34-s − 2·37-s − 40-s − 2·41-s − 8·43-s − 44-s + 4·46-s + 12·47-s + 50-s + 6·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.371·29-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.158·40-s − 0.312·41-s − 1.21·43-s − 0.150·44-s + 0.589·46-s + 1.75·47-s + 0.141·50-s + 0.832·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.613108016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.613108016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.65696433291723, −13.83672311177402, −13.62960807999824, −13.08960711133268, −12.50266211633508, −12.03823128371386, −11.55600853945414, −11.00359185837493, −10.52655400750030, −10.11546840513510, −9.231182258630260, −8.751209457047187, −8.138851202008714, −7.667933810101527, −7.059062504508958, −6.495722358596143, −5.801028156693002, −5.439414373945102, −4.786471833681900, −4.029588916523651, −3.466867788028849, −3.179228526499256, −2.224933140126235, −1.353138483527294, −0.7335129984573424,
0.7335129984573424, 1.353138483527294, 2.224933140126235, 3.179228526499256, 3.466867788028849, 4.029588916523651, 4.786471833681900, 5.439414373945102, 5.801028156693002, 6.495722358596143, 7.059062504508958, 7.667933810101527, 8.138851202008714, 8.751209457047187, 9.231182258630260, 10.11546840513510, 10.52655400750030, 11.00359185837493, 11.55600853945414, 12.03823128371386, 12.50266211633508, 13.08960711133268, 13.62960807999824, 13.83672311177402, 14.65696433291723
