| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 7-s + 9-s − 5·11-s + 2·12-s + 13-s + 2·14-s − 4·16-s − 2·18-s − 6·19-s − 21-s + 10·22-s + 3·23-s − 2·26-s + 27-s − 2·28-s − 2·29-s − 2·31-s + 8·32-s − 5·33-s + 2·36-s + 7·37-s + 12·38-s + 39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 0.277·13-s + 0.534·14-s − 16-s − 0.471·18-s − 1.37·19-s − 0.218·21-s + 2.13·22-s + 0.625·23-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.359·31-s + 1.41·32-s − 0.870·33-s + 1/3·36-s + 1.15·37-s + 1.94·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−13.12677969533865, −12.59574454816701, −12.04565464941278, −11.14195820716384, −11.05094239647208, −10.58685865588309, −10.08464279879917, −9.701792744541333, −9.312695606967501, −8.663502747913812, −8.497378025929014, −7.856539657379730, −7.692878610959794, −7.082530738762080, −6.533528909344967, −6.136766591625867, −5.384705344500091, −4.770225689549434, −4.389443711394286, −3.618947082659860, −2.981946251457358, −2.568816018817765, −1.920902615297584, −1.486993528314810, −0.5539898100374472, 0,
0.5539898100374472, 1.486993528314810, 1.920902615297584, 2.568816018817765, 2.981946251457358, 3.618947082659860, 4.389443711394286, 4.770225689549434, 5.384705344500091, 6.136766591625867, 6.533528909344967, 7.082530738762080, 7.692878610959794, 7.856539657379730, 8.497378025929014, 8.663502747913812, 9.312695606967501, 9.701792744541333, 10.08464279879917, 10.58685865588309, 11.05094239647208, 11.14195820716384, 12.04565464941278, 12.59574454816701, 13.12677969533865
