| L(s) = 1 | + 3-s + 9-s + 11-s − 4·13-s − 8·19-s − 4·23-s + 27-s + 2·29-s + 4·31-s + 33-s − 8·37-s − 4·39-s − 2·41-s + 4·47-s − 7·49-s − 12·53-s − 8·57-s − 12·59-s + 2·61-s − 12·67-s − 4·69-s + 12·73-s + 12·79-s + 81-s + 8·83-s + 2·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.83·19-s − 0.834·23-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 1.31·37-s − 0.640·39-s − 0.312·41-s + 0.583·47-s − 49-s − 1.64·53-s − 1.05·57-s − 1.56·59-s + 0.256·61-s − 1.46·67-s − 0.481·69-s + 1.40·73-s + 1.35·79-s + 1/9·81-s + 0.878·83-s + 0.214·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−8.212252740681163083998696209774, −7.70142060108053211805964222156, −6.69590875784698134371445393548, −6.24067700591786084059293884607, −4.99233977340108707820670322738, −4.39774275121613848757969205639, −3.47991502532193488834025010823, −2.48980545019387184950215452816, −1.71378477981861071069940594747, 0,
1.71378477981861071069940594747, 2.48980545019387184950215452816, 3.47991502532193488834025010823, 4.39774275121613848757969205639, 4.99233977340108707820670322738, 6.24067700591786084059293884607, 6.69590875784698134371445393548, 7.70142060108053211805964222156, 8.212252740681163083998696209774
