| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 3·17-s + 4·19-s + 2·21-s + 9·23-s − 5·25-s + 5·27-s − 3·29-s − 4·31-s + 4·37-s − 6·41-s − 43-s − 3·49-s + 3·51-s + 9·53-s − 4·57-s + 6·59-s − 5·61-s + 4·63-s + 2·67-s − 9·69-s − 12·71-s − 10·73-s + 5·75-s − 13·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.727·17-s + 0.917·19-s + 0.436·21-s + 1.87·23-s − 25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.657·37-s − 0.937·41-s − 0.152·43-s − 3/7·49-s + 0.420·51-s + 1.23·53-s − 0.529·57-s + 0.781·59-s − 0.640·61-s + 0.503·63-s + 0.244·67-s − 1.08·69-s − 1.42·71-s − 1.17·73-s + 0.577·75-s − 1.46·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−12.92928498750775, −12.35620719925236, −11.76226244229817, −11.43922619077654, −11.23127435678162, −10.56252264065255, −10.16216031612184, −9.624006598347088, −9.213759393919021, −8.695258747606854, −8.454208897753328, −7.531446896861810, −7.255245155499225, −6.802101285029080, −6.252929888107900, −5.723168580948424, −5.466669508257472, −4.871968003146089, −4.332913653395098, −3.694338030473786, −3.025566385982801, −2.893646021981374, −2.020144821067413, −1.350882945318462, −0.5911302968396182, 0,
0.5911302968396182, 1.350882945318462, 2.020144821067413, 2.893646021981374, 3.025566385982801, 3.694338030473786, 4.332913653395098, 4.871968003146089, 5.466669508257472, 5.723168580948424, 6.252929888107900, 6.802101285029080, 7.255245155499225, 7.531446896861810, 8.454208897753328, 8.695258747606854, 9.213759393919021, 9.624006598347088, 10.16216031612184, 10.56252264065255, 11.23127435678162, 11.43922619077654, 11.76226244229817, 12.35620719925236, 12.92928498750775
