| L(s) = 1 | − 4·7-s − 4·11-s + 13-s + 4·17-s + 6·19-s − 5·25-s − 6·29-s + 4·31-s + 6·37-s − 6·41-s − 4·43-s + 8·47-s + 9·49-s + 6·53-s − 12·59-s + 10·61-s − 2·67-s + 12·71-s − 10·73-s + 16·77-s − 10·79-s − 4·83-s + 10·89-s − 4·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s + 1.37·19-s − 25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.244·67-s + 1.42·71-s − 1.17·73-s + 1.82·77-s − 1.12·79-s − 0.439·83-s + 1.05·89-s − 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−16.19142510940455, −15.80941546060828, −15.46108627283709, −14.72599051244762, −13.95272685462702, −13.34258554015124, −13.21132837111455, −12.43620365332295, −11.90594063108662, −11.35102846145944, −10.45476196383839, −10.03573023712802, −9.625322251124933, −9.038387825204136, −8.134206435233118, −7.617781200213570, −7.107059617069757, −6.273628698658621, −5.663683948914555, −5.298261741697956, −4.226542561895984, −3.395601039290671, −3.058628778241201, −2.190198983630017, −0.9761159537660773, 0,
0.9761159537660773, 2.190198983630017, 3.058628778241201, 3.395601039290671, 4.226542561895984, 5.298261741697956, 5.663683948914555, 6.273628698658621, 7.107059617069757, 7.617781200213570, 8.134206435233118, 9.038387825204136, 9.625322251124933, 10.03573023712802, 10.45476196383839, 11.35102846145944, 11.90594063108662, 12.43620365332295, 13.21132837111455, 13.34258554015124, 13.95272685462702, 14.72599051244762, 15.46108627283709, 15.80941546060828, 16.19142510940455
