| L(s) = 1 | + 7-s + 4·11-s + 6·13-s − 6·19-s − 23-s − 5·25-s − 3·29-s − 3·37-s − 6·41-s − 7·43-s − 12·47-s + 49-s + 3·53-s + 10·59-s − 2·61-s − 4·67-s − 3·71-s + 8·73-s + 4·77-s + 7·79-s + 6·83-s + 10·89-s + 6·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 1.66·13-s − 1.37·19-s − 0.208·23-s − 25-s − 0.557·29-s − 0.493·37-s − 0.937·41-s − 1.06·43-s − 1.75·47-s + 1/7·49-s + 0.412·53-s + 1.30·59-s − 0.256·61-s − 0.488·67-s − 0.356·71-s + 0.936·73-s + 0.455·77-s + 0.787·79-s + 0.658·83-s + 1.05·89-s + 0.628·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.43574068792155, −13.66122239903335, −13.46966590611814, −12.99583561430551, −12.21691342577250, −11.82760663895133, −11.29751946003073, −10.99645792108384, −10.30685946257512, −9.877038544872469, −9.116612476394839, −8.776477897183930, −8.234733414194977, −7.905904901525256, −6.944214904869175, −6.533159725064554, −6.164728087873190, −5.530658324948443, −4.827532504228898, −4.166645696413697, −3.661029056847679, −3.337871638593400, −2.063248919383287, −1.775246415766502, −1.031723968515204, 0,
1.031723968515204, 1.775246415766502, 2.063248919383287, 3.337871638593400, 3.661029056847679, 4.166645696413697, 4.827532504228898, 5.530658324948443, 6.164728087873190, 6.533159725064554, 6.944214904869175, 7.905904901525256, 8.234733414194977, 8.776477897183930, 9.116612476394839, 9.877038544872469, 10.30685946257512, 10.99645792108384, 11.29751946003073, 11.82760663895133, 12.21691342577250, 12.99583561430551, 13.46966590611814, 13.66122239903335, 14.43574068792155
