| L(s) = 1 | − 3-s + 5-s + 9-s + 11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 25-s − 27-s + 2·29-s + 8·31-s − 33-s − 2·37-s − 2·39-s + 10·41-s − 12·43-s + 45-s − 2·51-s + 2·53-s + 55-s + 4·57-s + 6·61-s + 2·65-s − 8·67-s − 8·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 1.56·41-s − 1.82·43-s + 0.149·45-s − 0.280·51-s + 0.274·53-s + 0.134·55-s + 0.529·57-s + 0.768·61-s + 0.248·65-s − 0.977·67-s − 0.949·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.55160307807570, −13.87679872047511, −13.43176472165983, −13.03800339106506, −12.39564558500952, −11.93130968573829, −11.56566698029174, −10.72082927083595, −10.63202383641639, −9.881224144227269, −9.547910706940096, −8.785415147921908, −8.349651022507967, −7.833723267933731, −6.997085186376213, −6.598693699401626, −6.098325946229670, −5.616854253677544, −4.966476943363612, −4.364828906440225, −3.854593972443804, −3.023993712272886, −2.412558189574237, −1.525864088985799, −1.023512521918839, 0,
1.023512521918839, 1.525864088985799, 2.412558189574237, 3.023993712272886, 3.854593972443804, 4.364828906440225, 4.966476943363612, 5.616854253677544, 6.098325946229670, 6.598693699401626, 6.997085186376213, 7.833723267933731, 8.349651022507967, 8.785415147921908, 9.547910706940096, 9.881224144227269, 10.63202383641639, 10.72082927083595, 11.56566698029174, 11.93130968573829, 12.39564558500952, 13.03800339106506, 13.43176472165983, 13.87679872047511, 14.55160307807570
