| L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s + 4·10-s − 11-s + 13-s + 16-s − 17-s − 5·19-s − 4·20-s + 22-s + 5·23-s + 11·25-s − 26-s + 3·29-s − 32-s + 34-s + 5·37-s + 5·38-s + 4·40-s − 6·41-s − 9·43-s − 44-s − 5·46-s − 3·47-s − 11·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.242·17-s − 1.14·19-s − 0.894·20-s + 0.213·22-s + 1.04·23-s + 11/5·25-s − 0.196·26-s + 0.557·29-s − 0.176·32-s + 0.171·34-s + 0.821·37-s + 0.811·38-s + 0.632·40-s − 0.937·41-s − 1.37·43-s − 0.150·44-s − 0.737·46-s − 0.437·47-s − 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.12744830568520, −13.36579601412159, −12.86568159990667, −12.48121511512652, −11.96523125696978, −11.50692340488966, −10.96358713216450, −10.84538962571656, −10.23298511481113, −9.513251529532015, −9.036871870609495, −8.444087334265363, −8.129038423783744, −7.823607490496824, −7.150246430439250, −6.608242358087846, −6.404742290126892, −5.388878444174093, −4.773424179172722, −4.357645496721459, −3.747571799296394, −3.046119895717920, −2.788903127978959, −1.705121245131327, −1.109855389618476, 0, 0,
1.109855389618476, 1.705121245131327, 2.788903127978959, 3.046119895717920, 3.747571799296394, 4.357645496721459, 4.773424179172722, 5.388878444174093, 6.404742290126892, 6.608242358087846, 7.150246430439250, 7.823607490496824, 8.129038423783744, 8.444087334265363, 9.036871870609495, 9.513251529532015, 10.23298511481113, 10.84538962571656, 10.96358713216450, 11.50692340488966, 11.96523125696978, 12.48121511512652, 12.86568159990667, 13.36579601412159, 14.12744830568520
