| L(s) = 1 | − 3·5-s − 4·7-s − 5·11-s + 2·13-s + 5·19-s + 5·23-s + 4·25-s − 6·31-s + 12·35-s + 9·37-s − 6·41-s + 4·43-s + 12·47-s + 9·49-s + 5·53-s + 15·55-s − 7·59-s + 5·61-s − 6·65-s − 67-s + 4·71-s + 20·77-s + 12·79-s + 83-s − 3·89-s − 8·91-s − 15·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.51·7-s − 1.50·11-s + 0.554·13-s + 1.14·19-s + 1.04·23-s + 4/5·25-s − 1.07·31-s + 2.02·35-s + 1.47·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.686·53-s + 2.02·55-s − 0.911·59-s + 0.640·61-s − 0.744·65-s − 0.122·67-s + 0.474·71-s + 2.27·77-s + 1.35·79-s + 0.109·83-s − 0.317·89-s − 0.838·91-s − 1.53·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7687758885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7687758885\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.84912183281552, −13.85023966829047, −13.44458151522928, −13.05618611809471, −12.43955128748734, −12.20171954443011, −11.42400889305287, −10.88724292562323, −10.62535316684225, −9.788430900882363, −9.421434575585499, −8.791537854604842, −8.167871641266334, −7.593354976766421, −7.257218028803564, −6.708799860279062, −5.883299038274587, −5.443451356336261, −4.766961282054830, −3.912935942199822, −3.558422027721426, −2.906043720780158, −2.484626434362845, −1.065498263238466, −0.3634366819957898,
0.3634366819957898, 1.065498263238466, 2.484626434362845, 2.906043720780158, 3.558422027721426, 3.912935942199822, 4.766961282054830, 5.443451356336261, 5.883299038274587, 6.708799860279062, 7.257218028803564, 7.593354976766421, 8.167871641266334, 8.791537854604842, 9.421434575585499, 9.788430900882363, 10.62535316684225, 10.88724292562323, 11.42400889305287, 12.20171954443011, 12.43955128748734, 13.05618611809471, 13.44458151522928, 13.85023966829047, 14.84912183281552
