| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 2·11-s − 13-s − 17-s − 19-s − 3·21-s + 6·23-s − 9·27-s − 29-s − 5·31-s + 6·33-s − 2·37-s + 3·39-s − 10·41-s − 8·43-s + 11·47-s + 49-s + 3·51-s + 7·53-s + 3·57-s − 9·59-s + 7·61-s + 6·63-s − 8·67-s − 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 0.603·11-s − 0.277·13-s − 0.242·17-s − 0.229·19-s − 0.654·21-s + 1.25·23-s − 1.73·27-s − 0.185·29-s − 0.898·31-s + 1.04·33-s − 0.328·37-s + 0.480·39-s − 1.56·41-s − 1.21·43-s + 1.60·47-s + 1/7·49-s + 0.420·51-s + 0.961·53-s + 0.397·57-s − 1.17·59-s + 0.896·61-s + 0.755·63-s − 0.977·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−13.37870163309837, −13.02292472835391, −12.54432365239941, −12.08488181868233, −11.63979555005800, −11.29215034247175, −10.78626478258921, −10.36456770415811, −10.15807170419425, −9.385909425402597, −8.842052989980729, −8.388627836773766, −7.611388437886180, −7.166673000623471, −6.865424402117083, −6.287400369870416, −5.648159456934302, −5.266684733266343, −5.015112213525091, −4.346983438402300, −3.886451473693572, −3.039124099261775, −2.399894249824524, −1.527543057407315, −1.174543719853865, 0, 0,
1.174543719853865, 1.527543057407315, 2.399894249824524, 3.039124099261775, 3.886451473693572, 4.346983438402300, 5.015112213525091, 5.266684733266343, 5.648159456934302, 6.287400369870416, 6.865424402117083, 7.166673000623471, 7.611388437886180, 8.388627836773766, 8.842052989980729, 9.385909425402597, 10.15807170419425, 10.36456770415811, 10.78626478258921, 11.29215034247175, 11.63979555005800, 12.08488181868233, 12.54432365239941, 13.02292472835391, 13.37870163309837
