| L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 2·26-s + 28-s − 5·32-s + 6·34-s − 2·35-s − 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s − 4·44-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.883·32-s + 1.02·34-s − 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8195184391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8195184391\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.35105028040516, −13.87131600407085, −13.54081495429899, −12.99484755192615, −12.43915589679279, −12.01238849825476, −11.06682634469997, −10.82019075961379, −10.22728882024451, −9.500065155456359, −9.380310759754133, −8.941603203205257, −8.332043807860524, −7.759753555159382, −6.944186881625877, −6.579748248069606, −6.088028520538525, −5.305976908528458, −4.684854095635469, −4.146060445814317, −3.602042915769110, −2.536819246060319, −1.959174269454554, −1.376268863456180, −0.3614178923041300,
0.3614178923041300, 1.376268863456180, 1.959174269454554, 2.536819246060319, 3.602042915769110, 4.146060445814317, 4.684854095635469, 5.305976908528458, 6.088028520538525, 6.579748248069606, 6.944186881625877, 7.759753555159382, 8.332043807860524, 8.941603203205257, 9.380310759754133, 9.500065155456359, 10.22728882024451, 10.82019075961379, 11.06682634469997, 12.01238849825476, 12.43915589679279, 12.99484755192615, 13.54081495429899, 13.87131600407085, 14.35105028040516
