| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 2·11-s + 6·13-s + 14-s + 16-s − 17-s − 4·19-s + 20-s − 2·22-s + 4·23-s + 25-s − 6·26-s − 28-s + 10·29-s − 32-s + 34-s − 35-s + 12·37-s + 4·38-s − 40-s + 6·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 1.85·29-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 1.97·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926644832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926644832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−16.52953118835560, −16.14976482426960, −15.50080621307238, −14.94859476451983, −14.26176050955138, −13.68677012448713, −12.94128184922441, −12.72008599543673, −11.65100723562102, −11.27113066240901, −10.66023964058658, −10.08745507732870, −9.424674123424087, −8.847026752445764, −8.416952685819067, −7.747966331826732, −6.653339202057954, −6.445876281355992, −5.932469297168671, −4.844722048239507, −4.086561287676270, −3.235032836918034, −2.500764532914791, −1.468990600130451, −0.7877732800705842,
0.7877732800705842, 1.468990600130451, 2.500764532914791, 3.235032836918034, 4.086561287676270, 4.844722048239507, 5.932469297168671, 6.445876281355992, 6.653339202057954, 7.747966331826732, 8.416952685819067, 8.847026752445764, 9.424674123424087, 10.08745507732870, 10.66023964058658, 11.27113066240901, 11.65100723562102, 12.72008599543673, 12.94128184922441, 13.68677012448713, 14.26176050955138, 14.94859476451983, 15.50080621307238, 16.14976482426960, 16.52953118835560
