| L(s) = 1 | + 2·5-s − 7-s + 4·11-s + 13-s − 2·17-s + 4·19-s + 4·23-s − 25-s + 2·29-s − 2·35-s + 2·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s + 8·55-s + 4·59-s + 14·61-s + 2·65-s + 12·67-s − 8·71-s + 14·73-s − 4·77-s − 12·79-s − 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.07·55-s + 0.520·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.455·77-s − 1.35·79-s − 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.379174022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.379174022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.27922306280647, −14.71096009173521, −14.08422125368467, −13.81589864346610, −13.23706461535073, −12.66533673522536, −12.16123278207408, −11.36260648578959, −11.18130134917784, −10.26265410963787, −9.803685157726860, −9.304745700413721, −8.900469395032588, −8.245357246200463, −7.396746744265283, −6.756885262668195, −6.466965092858149, −5.576424973960139, −5.363922555410871, −4.231900662913383, −3.875253509686349, −2.929948314001243, −2.336479250263135, −1.415483235012565, −0.7846426241452845,
0.7846426241452845, 1.415483235012565, 2.336479250263135, 2.929948314001243, 3.875253509686349, 4.231900662913383, 5.363922555410871, 5.576424973960139, 6.466965092858149, 6.756885262668195, 7.396746744265283, 8.245357246200463, 8.900469395032588, 9.304745700413721, 9.803685157726860, 10.26265410963787, 11.18130134917784, 11.36260648578959, 12.16123278207408, 12.66533673522536, 13.23706461535073, 13.81589864346610, 14.08422125368467, 14.71096009173521, 15.27922306280647
