| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 2·7-s − 8-s + 9-s − 3·10-s + 2·11-s − 12-s + 13-s + 2·14-s − 3·15-s + 16-s + 5·17-s − 18-s − 19-s + 3·20-s + 2·21-s − 2·22-s + 6·23-s + 24-s + 4·25-s − 26-s − 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.774·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.229·19-s + 0.670·20-s + 0.436·21-s − 0.426·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 246 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9386205914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9386205914\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.03692258948577384580923570419, −10.89373506595971332057904971194, −9.953409099042967750861409917190, −9.534418094728046955756604334195, −8.356647240792238542821126916269, −6.78430324901641246819336686633, −6.26291645490747885242862369082, −5.13341372798534434135976920906, −3.11920280674024255742590168743, −1.37221072992549302129653289340,
1.37221072992549302129653289340, 3.11920280674024255742590168743, 5.13341372798534434135976920906, 6.26291645490747885242862369082, 6.78430324901641246819336686633, 8.356647240792238542821126916269, 9.534418094728046955756604334195, 9.953409099042967750861409917190, 10.89373506595971332057904971194, 12.03692258948577384580923570419
