| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 4·7-s − 8-s − 2·9-s + 4·10-s − 6·11-s − 12-s + 13-s + 4·14-s + 4·15-s + 16-s − 3·17-s + 2·18-s + 2·19-s − 4·20-s + 4·21-s + 6·22-s + 3·23-s + 24-s + 11·25-s − 26-s + 5·27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s − 0.894·20-s + 0.872·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.962·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.40160515973158, −15.11706515462648, −14.26696175366458, −13.36246390310274, −12.94693028761415, −12.62047575157993, −11.92232617849996, −11.49857714246804, −11.02121548179073, −10.52331154426319, −10.23342777522262, −9.208046685619589, −8.898495114118938, −8.348545254263628, −7.640182419110242, −7.288528877776439, −6.859730180414440, −6.041675026004733, −5.517544737847075, −4.932544101181687, −4.028327651126600, −3.402888240712043, −2.951347450038229, −2.330963451275381, −0.8401139432202426, 0, 0,
0.8401139432202426, 2.330963451275381, 2.951347450038229, 3.402888240712043, 4.028327651126600, 4.932544101181687, 5.517544737847075, 6.041675026004733, 6.859730180414440, 7.288528877776439, 7.640182419110242, 8.348545254263628, 8.898495114118938, 9.208046685619589, 10.23342777522262, 10.52331154426319, 11.02121548179073, 11.49857714246804, 11.92232617849996, 12.62047575157993, 12.94693028761415, 13.36246390310274, 14.26696175366458, 15.11706515462648, 15.40160515973158
