| L(s) = 1 | + 3·5-s + 2·7-s − 5·11-s + 7·13-s + 17-s + 7·19-s + 4·23-s + 4·25-s − 8·29-s + 31-s + 6·35-s + 6·37-s + 2·41-s − 10·43-s − 47-s − 3·49-s + 6·53-s − 15·55-s + 10·59-s − 61-s + 21·65-s − 3·67-s + 3·71-s + 14·73-s − 10·77-s + 11·79-s − 7·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.50·11-s + 1.94·13-s + 0.242·17-s + 1.60·19-s + 0.834·23-s + 4/5·25-s − 1.48·29-s + 0.179·31-s + 1.01·35-s + 0.986·37-s + 0.312·41-s − 1.52·43-s − 0.145·47-s − 3/7·49-s + 0.824·53-s − 2.02·55-s + 1.30·59-s − 0.128·61-s + 2.60·65-s − 0.366·67-s + 0.356·71-s + 1.63·73-s − 1.13·77-s + 1.23·79-s − 0.768·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.861562763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.861562763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−15.90268579364293, −15.20231085707225, −14.74835670109591, −13.95774689802717, −13.56461080339361, −13.19754814662168, −12.84413385012759, −11.75665761470492, −11.25797665587852, −10.81419857772755, −10.24774778938536, −9.579356841610920, −9.181383689787065, −8.294127783257201, −7.992630088618274, −7.225752931339057, −6.456181898678322, −5.719005693511871, −5.384753809117200, −4.926242148756614, −3.780216669321274, −3.128835665959994, −2.313157516778151, −1.563968282275312, −0.8945673214318048,
0.8945673214318048, 1.563968282275312, 2.313157516778151, 3.128835665959994, 3.780216669321274, 4.926242148756614, 5.384753809117200, 5.719005693511871, 6.456181898678322, 7.225752931339057, 7.992630088618274, 8.294127783257201, 9.181383689787065, 9.579356841610920, 10.24774778938536, 10.81419857772755, 11.25797665587852, 11.75665761470492, 12.84413385012759, 13.19754814662168, 13.56461080339361, 13.95774689802717, 14.74835670109591, 15.20231085707225, 15.90268579364293
