| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s + 17-s − 19-s − 21-s − 5·23-s − 27-s + 6·29-s − 6·31-s − 33-s + 9·37-s − 5·41-s + 4·43-s − 7·47-s − 6·49-s − 51-s − 4·53-s + 57-s − 9·59-s + 12·61-s + 63-s + 4·67-s + 5·69-s + 15·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.242·17-s − 0.229·19-s − 0.218·21-s − 1.04·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.174·33-s + 1.47·37-s − 0.780·41-s + 0.609·43-s − 1.02·47-s − 6/7·49-s − 0.140·51-s − 0.549·53-s + 0.132·57-s − 1.17·59-s + 1.53·61-s + 0.125·63-s + 0.488·67-s + 0.601·69-s + 1.78·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.59475132533716, −14.29614740163529, −13.76439447092745, −13.03563106900558, −12.65972252133609, −12.15032435422108, −11.56861290792117, −11.19580368852999, −10.73087375263945, −9.939831222550614, −9.764544275695851, −9.037525131070812, −8.309910185875744, −7.966753029469210, −7.361352089570598, −6.569174355354177, −6.329032532368488, −5.597533248116726, −5.056056151748506, −4.466820365613496, −3.903422535657871, −3.220524378779122, −2.350340341372709, −1.683009193379275, −0.9389493477724894, 0,
0.9389493477724894, 1.683009193379275, 2.350340341372709, 3.220524378779122, 3.903422535657871, 4.466820365613496, 5.056056151748506, 5.597533248116726, 6.329032532368488, 6.569174355354177, 7.361352089570598, 7.966753029469210, 8.309910185875744, 9.037525131070812, 9.764544275695851, 9.939831222550614, 10.73087375263945, 11.19580368852999, 11.56861290792117, 12.15032435422108, 12.65972252133609, 13.03563106900558, 13.76439447092745, 14.29614740163529, 14.59475132533716
