| L(s) = 1 | − 2·5-s − 3·7-s + 3·11-s + 6·13-s − 5·17-s − 2·19-s + 4·23-s − 25-s − 7·29-s + 5·31-s + 6·35-s + 11·37-s + 2·41-s + 8·43-s + 2·49-s + 6·53-s − 6·55-s + 5·59-s − 5·61-s − 12·65-s + 2·67-s − 2·71-s − 9·77-s + 14·79-s − 83-s + 10·85-s − 18·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 0.904·11-s + 1.66·13-s − 1.21·17-s − 0.458·19-s + 0.834·23-s − 1/5·25-s − 1.29·29-s + 0.898·31-s + 1.01·35-s + 1.80·37-s + 0.312·41-s + 1.21·43-s + 2/7·49-s + 0.824·53-s − 0.809·55-s + 0.650·59-s − 0.640·61-s − 1.48·65-s + 0.244·67-s − 0.237·71-s − 1.02·77-s + 1.57·79-s − 0.109·83-s + 1.08·85-s − 1.88·91-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
| \(L(1)\) |
\(\approx\) |
\(1.598017966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598017966\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.82642878596650, −13.90718882296845, −13.42267791569316, −13.10618422673604, −12.59608718917382, −11.97433538531779, −11.31493298168470, −11.12059004719938, −10.61876561226468, −9.744222613440166, −9.193812412176075, −8.971451544394048, −8.246295435511844, −7.767309099178384, −6.960939765868134, −6.571578168164123, −6.111397611440806, −5.570373737319748, −4.408499570713411, −4.107428548257359, −3.653108861532454, −2.973499353378006, −2.217912269466605, −1.186871175072455, −0.5022317901430736,
0.5022317901430736, 1.186871175072455, 2.217912269466605, 2.973499353378006, 3.653108861532454, 4.107428548257359, 4.408499570713411, 5.570373737319748, 6.111397611440806, 6.571578168164123, 6.960939765868134, 7.767309099178384, 8.246295435511844, 8.971451544394048, 9.193812412176075, 9.744222613440166, 10.61876561226468, 11.12059004719938, 11.31493298168470, 11.97433538531779, 12.59608718917382, 13.10618422673604, 13.42267791569316, 13.90718882296845, 14.82642878596650
