| L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s + 11-s + 3·14-s + 16-s − 2·17-s + 2·19-s + 20-s + 22-s − 3·23-s + 25-s + 3·28-s − 29-s + 7·31-s + 32-s − 2·34-s + 3·35-s − 2·37-s + 2·38-s + 40-s − 5·41-s + 11·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.566·28-s − 0.185·29-s + 1.25·31-s + 0.176·32-s − 0.342·34-s + 0.507·35-s − 0.328·37-s + 0.324·38-s + 0.158·40-s − 0.780·41-s + 1.67·43-s + 0.150·44-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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.54223584174045, −13.11123506542325, −12.48068593930894, −12.05509786293058, −11.66851467614950, −11.06092541325689, −10.89869641015223, −10.20097762631866, −9.656290475342230, −9.292412442819183, −8.504533086192322, −8.146281658533175, −7.710520542287476, −7.089934644031375, −6.409513407508181, −6.265748154375243, −5.374622114136383, −5.147824070796210, −4.522755391210730, −4.130756627366123, −3.443203920642577, −2.765297309647260, −2.243633309044562, −1.537444116534395, −1.194535679140023, 0,
1.194535679140023, 1.537444116534395, 2.243633309044562, 2.765297309647260, 3.443203920642577, 4.130756627366123, 4.522755391210730, 5.147824070796210, 5.374622114136383, 6.265748154375243, 6.409513407508181, 7.089934644031375, 7.710520542287476, 8.146281658533175, 8.504533086192322, 9.292412442819183, 9.656290475342230, 10.20097762631866, 10.89869641015223, 11.06092541325689, 11.66851467614950, 12.05509786293058, 12.48068593930894, 13.11123506542325, 13.54223584174045
