| L(s) = 1 | − 3-s + 9-s + 4·11-s − 6·13-s − 4·17-s − 2·19-s + 23-s − 5·25-s − 27-s − 2·29-s + 4·31-s − 4·33-s − 2·37-s + 6·39-s − 2·41-s + 10·43-s + 4·51-s + 12·53-s + 2·57-s + 12·59-s − 6·61-s − 10·67-s − 69-s − 8·71-s + 14·73-s + 5·75-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.970·17-s − 0.458·19-s + 0.208·23-s − 25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s + 1.52·43-s + 0.560·51-s + 1.64·53-s + 0.264·57-s + 1.56·59-s − 0.768·61-s − 1.22·67-s − 0.120·69-s − 0.949·71-s + 1.63·73-s + 0.577·75-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285988330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285988330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.92375528971627, −12.41712089678371, −11.94812266115691, −11.70067131564559, −11.26390533473448, −10.62652757639114, −10.16519268520743, −9.795471949935833, −9.206761774452191, −8.878330793404986, −8.327863886929912, −7.523192483272200, −7.248427238090721, −6.817782923125126, −6.206625128549343, −5.843728521407886, −5.212264604649000, −4.598395633128568, −4.279856452755583, −3.782505998032915, −2.989965583912603, −2.214576904799765, −1.983533452484873, −1.030033848636185, −0.3647971740408150,
0.3647971740408150, 1.030033848636185, 1.983533452484873, 2.214576904799765, 2.989965583912603, 3.782505998032915, 4.279856452755583, 4.598395633128568, 5.212264604649000, 5.843728521407886, 6.206625128549343, 6.817782923125126, 7.248427238090721, 7.523192483272200, 8.327863886929912, 8.878330793404986, 9.206761774452191, 9.795471949935833, 10.16519268520743, 10.62652757639114, 11.26390533473448, 11.70067131564559, 11.94812266115691, 12.41712089678371, 12.92375528971627
