| L(s) = 1 | − 3·7-s + 2·11-s − 3·13-s − 19-s + 23-s − 31-s + 6·37-s − 4·41-s + 5·43-s + 10·47-s + 2·49-s − 4·53-s − 6·59-s + 61-s − 7·67-s − 4·71-s + 2·73-s − 6·77-s + 14·83-s + 2·89-s + 9·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.603·11-s − 0.832·13-s − 0.229·19-s + 0.208·23-s − 0.179·31-s + 0.986·37-s − 0.624·41-s + 0.762·43-s + 1.45·47-s + 2/7·49-s − 0.549·53-s − 0.781·59-s + 0.128·61-s − 0.855·67-s − 0.474·71-s + 0.234·73-s − 0.683·77-s + 1.53·83-s + 0.211·89-s + 0.943·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−14.25501009886729, −13.46725585463485, −13.39715125969518, −12.62486102733714, −12.18465824555676, −12.00673475797133, −11.09894618629521, −10.78341190558973, −10.11148505884878, −9.652705795709259, −9.246788901956346, −8.853171483266747, −8.087795244491725, −7.494770335354868, −7.085067386597513, −6.414311180817720, −6.133277463767044, −5.440157928094211, −4.784398185797733, −4.169758108150684, −3.654268272107425, −2.932144755781320, −2.506823846101119, −1.664215575381839, −0.7922580600373709, 0,
0.7922580600373709, 1.664215575381839, 2.506823846101119, 2.932144755781320, 3.654268272107425, 4.169758108150684, 4.784398185797733, 5.440157928094211, 6.133277463767044, 6.414311180817720, 7.085067386597513, 7.494770335354868, 8.087795244491725, 8.853171483266747, 9.246788901956346, 9.652705795709259, 10.11148505884878, 10.78341190558973, 11.09894618629521, 12.00673475797133, 12.18465824555676, 12.62486102733714, 13.39715125969518, 13.46725585463485, 14.25501009886729
