| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 3·11-s + 6·13-s + 3·15-s + 7·17-s + 3·19-s − 4·23-s + 4·25-s − 5·27-s − 6·29-s + 4·31-s + 3·33-s + 2·37-s + 6·39-s − 6·41-s − 7·43-s − 6·45-s + 6·47-s + 7·51-s − 10·53-s + 9·55-s + 3·57-s + 4·59-s − 5·61-s + 18·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.904·11-s + 1.66·13-s + 0.774·15-s + 1.69·17-s + 0.688·19-s − 0.834·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 1.06·43-s − 0.894·45-s + 0.875·47-s + 0.980·51-s − 1.37·53-s + 1.21·55-s + 0.397·57-s + 0.520·59-s − 0.640·61-s + 2.23·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.165319942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.165319942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 229 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.26433162535351, −12.93196810799288, −12.12439069293868, −11.76581289247829, −11.35398196545645, −10.74688629036914, −10.16360576317248, −9.791224248316127, −9.350003429606506, −8.962414972520663, −8.460738416121698, −7.898426750420257, −7.608580993703160, −6.622040694396672, −6.254593039241817, −5.950500566665445, −5.369972387025686, −5.015276107181209, −3.892358319308495, −3.591261961819793, −3.198961982681394, −2.425410645081868, −1.762443990321404, −1.383000311796983, −0.6979394958741172,
0.6979394958741172, 1.383000311796983, 1.762443990321404, 2.425410645081868, 3.198961982681394, 3.591261961819793, 3.892358319308495, 5.015276107181209, 5.369972387025686, 5.950500566665445, 6.254593039241817, 6.622040694396672, 7.608580993703160, 7.898426750420257, 8.460738416121698, 8.962414972520663, 9.350003429606506, 9.791224248316127, 10.16360576317248, 10.74688629036914, 11.35398196545645, 11.76581289247829, 12.12439069293868, 12.93196810799288, 13.26433162535351
