| L(s) = 1 | − 2·5-s − 3·7-s − 5·11-s + 3·13-s + 3·17-s − 5·19-s + 3·23-s − 25-s + 4·31-s + 6·35-s − 37-s + 6·41-s − 4·43-s − 4·47-s + 2·49-s + 3·53-s + 10·55-s − 14·59-s + 14·61-s − 6·65-s − 12·67-s + 12·71-s + 13·73-s + 15·77-s + 6·79-s + 7·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s − 1.50·11-s + 0.832·13-s + 0.727·17-s − 1.14·19-s + 0.625·23-s − 1/5·25-s + 0.718·31-s + 1.01·35-s − 0.164·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s + 0.412·53-s + 1.34·55-s − 1.82·59-s + 1.79·61-s − 0.744·65-s − 1.46·67-s + 1.42·71-s + 1.52·73-s + 1.70·77-s + 0.675·79-s + 0.768·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.79428238124625, −15.36280081406218, −15.06282609272790, −14.19149097098898, −13.44166000599401, −13.20701306926739, −12.45236622057275, −12.32269983957633, −11.32296203944084, −10.94099682151727, −10.36023768675403, −9.838358326552860, −9.196071167657247, −8.380894260384363, −8.066987853340818, −7.480430679252932, −6.713051456207336, −6.224622235136119, −5.530148446990555, −4.825762917023059, −4.067636444563722, −3.409895319686237, −2.933887084555361, −2.088618502165905, −0.8012294822314576, 0,
0.8012294822314576, 2.088618502165905, 2.933887084555361, 3.409895319686237, 4.067636444563722, 4.825762917023059, 5.530148446990555, 6.224622235136119, 6.713051456207336, 7.480430679252932, 8.066987853340818, 8.380894260384363, 9.196071167657247, 9.838358326552860, 10.36023768675403, 10.94099682151727, 11.32296203944084, 12.32269983957633, 12.45236622057275, 13.20701306926739, 13.44166000599401, 14.19149097098898, 15.06282609272790, 15.36280081406218, 15.79428238124625
