| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 16-s − 2·17-s + 4·19-s − 20-s − 22-s + 25-s + 6·29-s + 4·31-s − 32-s + 2·34-s − 2·37-s − 4·38-s + 40-s + 6·41-s + 44-s − 12·47-s − 7·49-s − 50-s + 2·53-s − 55-s − 6·58-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 + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.150·44-s − 1.75·47-s − 49-s − 0.141·50-s + 0.274·53-s − 0.134·55-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.47059335951690, −12.91262407010204, −12.43159025914332, −11.88537731037252, −11.54975580044651, −11.16820413732250, −10.56624897732549, −10.10131098340676, −9.633203507239045, −9.206070934449547, −8.610833351693358, −8.214934633516426, −7.785631493718224, −7.220975426571829, −6.690673858740047, −6.355830128827372, −5.692713474308529, −5.013391078786411, −4.555994284817263, −3.938632684457239, −3.205244171017111, −2.868300543990606, −2.090818114577270, −1.374881434725782, −0.8095328510294384, 0,
0.8095328510294384, 1.374881434725782, 2.090818114577270, 2.868300543990606, 3.205244171017111, 3.938632684457239, 4.555994284817263, 5.013391078786411, 5.692713474308529, 6.355830128827372, 6.690673858740047, 7.220975426571829, 7.785631493718224, 8.214934633516426, 8.610833351693358, 9.206070934449547, 9.633203507239045, 10.10131098340676, 10.56624897732549, 11.16820413732250, 11.54975580044651, 11.88537731037252, 12.43159025914332, 12.91262407010204, 13.47059335951690
