| L(s) = 1 | + 3-s − 3·7-s − 2·9-s + 3·13-s + 3·17-s − 19-s − 3·21-s + 9·23-s − 5·25-s − 5·27-s − 9·29-s + 6·31-s + 6·37-s + 3·39-s − 6·41-s − 8·43-s + 2·49-s + 3·51-s − 9·53-s − 57-s − 3·59-s + 6·61-s + 6·63-s + 5·67-s + 9·69-s − 11·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.832·13-s + 0.727·17-s − 0.229·19-s − 0.654·21-s + 1.87·23-s − 25-s − 0.962·27-s − 1.67·29-s + 1.07·31-s + 0.986·37-s + 0.480·39-s − 0.937·41-s − 1.21·43-s + 2/7·49-s + 0.420·51-s − 1.23·53-s − 0.132·57-s − 0.390·59-s + 0.768·61-s + 0.755·63-s + 0.610·67-s + 1.08·69-s − 1.28·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.032426794852246132422935541717, −7.21572384406897604266002117601, −6.41135708570000694413697076800, −5.84751163699287195565665611317, −5.03693433302208422590208123380, −3.82799156115040825587137870484, −3.29504828689534582304511709282, −2.65892103233070842073651650226, −1.40981022497989024822456343202, 0,
1.40981022497989024822456343202, 2.65892103233070842073651650226, 3.29504828689534582304511709282, 3.82799156115040825587137870484, 5.03693433302208422590208123380, 5.84751163699287195565665611317, 6.41135708570000694413697076800, 7.21572384406897604266002117601, 8.032426794852246132422935541717
