| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s − 13-s − 16-s + 4·17-s + 2·19-s − 20-s − 4·23-s + 25-s + 26-s − 10·29-s + 2·31-s − 5·32-s − 4·34-s − 8·37-s − 2·38-s + 3·40-s + 6·41-s − 10·43-s + 4·46-s − 6·47-s − 50-s + 52-s − 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s − 0.277·13-s − 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 1.85·29-s + 0.359·31-s − 0.883·32-s − 0.685·34-s − 1.31·37-s − 0.324·38-s + 0.474·40-s + 0.937·41-s − 1.52·43-s + 0.589·46-s − 0.875·47-s − 0.141·50-s + 0.138·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8868570934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8868570934\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−15.12663327873125, −14.48990000333163, −14.24224824469563, −13.61363984257624, −13.03691810840769, −12.70737014930019, −11.85072789306072, −11.48139637346861, −10.63781359841721, −10.18586743992941, −9.738234041402643, −9.311801918450122, −8.719771942652739, −8.063464448513891, −7.635170571779596, −7.082620355581284, −6.272836190654688, −5.545729821873844, −5.150060024720762, −4.411403707478365, −3.648696791824027, −3.070183088069555, −1.896868405751286, −1.509959524309562, −0.4194876147148494,
0.4194876147148494, 1.509959524309562, 1.896868405751286, 3.070183088069555, 3.648696791824027, 4.411403707478365, 5.150060024720762, 5.545729821873844, 6.272836190654688, 7.082620355581284, 7.635170571779596, 8.063464448513891, 8.719771942652739, 9.311801918450122, 9.738234041402643, 10.18586743992941, 10.63781359841721, 11.48139637346861, 11.85072789306072, 12.70737014930019, 13.03691810840769, 13.61363984257624, 14.24224824469563, 14.48990000333163, 15.12663327873125
