| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 3·7-s + 8-s + 9-s + 3·10-s − 11-s − 12-s + 13-s − 3·14-s − 3·15-s + 16-s + 6·17-s + 18-s − 5·19-s + 3·20-s + 3·21-s − 22-s + 4·23-s − 24-s + 4·25-s + 26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.774·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.14·19-s + 0.670·20-s + 0.654·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.440623981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.440623981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 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
−14.50209772476959, −13.58427149145484, −13.27081439609698, −13.15171152087138, −12.38727509037505, −12.08643010572998, −11.42485266126976, −10.65158197635240, −10.32631104135653, −9.928709327268419, −9.478533946866005, −8.708695683325107, −8.198439226649304, −7.221811985437294, −6.788647553001321, −6.380069485993470, −5.806127152585097, −5.389025590167846, −4.984866243123900, −4.067295570890019, −3.403363304765425, −2.926800070653198, −2.107456787824475, −1.516068451209824, −0.5775312639581631,
0.5775312639581631, 1.516068451209824, 2.107456787824475, 2.926800070653198, 3.403363304765425, 4.067295570890019, 4.984866243123900, 5.389025590167846, 5.806127152585097, 6.380069485993470, 6.788647553001321, 7.221811985437294, 8.198439226649304, 8.708695683325107, 9.478533946866005, 9.928709327268419, 10.32631104135653, 10.65158197635240, 11.42485266126976, 12.08643010572998, 12.38727509037505, 13.15171152087138, 13.27081439609698, 13.58427149145484, 14.50209772476959
