| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s − 13-s + 16-s − 6·17-s + 6·19-s − 2·22-s + 4·23-s − 26-s − 4·31-s + 32-s − 6·34-s − 6·37-s + 6·38-s + 10·41-s + 8·43-s − 2·44-s + 4·46-s + 8·47-s − 52-s + 6·53-s + 8·59-s + 8·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.426·22-s + 0.834·23-s − 0.196·26-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.986·37-s + 0.973·38-s + 1.56·41-s + 1.21·43-s − 0.301·44-s + 0.589·46-s + 1.16·47-s − 0.138·52-s + 0.824·53-s + 1.04·59-s + 1.02·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.207955908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.207955908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.88763878237703, −12.24208381544413, −11.94028765772099, −11.25998789017219, −11.09744152077210, −10.45880921794693, −10.20029567406485, −9.357543033045951, −9.099882368085011, −8.654606587260959, −7.925227022638206, −7.376809395425832, −7.175397336800808, −6.681719569747381, −5.914097785016035, −5.585401620507324, −5.132579321317645, −4.586331686617259, −4.102050554999681, −3.571298219658855, −2.906228606558707, −2.459587199270431, −2.015454238234009, −1.113341088064929, −0.5119954021534653,
0.5119954021534653, 1.113341088064929, 2.015454238234009, 2.459587199270431, 2.906228606558707, 3.571298219658855, 4.102050554999681, 4.586331686617259, 5.132579321317645, 5.585401620507324, 5.914097785016035, 6.681719569747381, 7.175397336800808, 7.376809395425832, 7.925227022638206, 8.654606587260959, 9.099882368085011, 9.357543033045951, 10.20029567406485, 10.45880921794693, 11.09744152077210, 11.25998789017219, 11.94028765772099, 12.24208381544413, 12.88763878237703
