| L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s − 3·11-s − 2·13-s + 3·15-s + 2·19-s − 21-s + 4·23-s + 4·25-s + 27-s + 3·29-s + 3·31-s − 3·33-s − 3·35-s + 6·37-s − 2·39-s + 4·41-s − 10·43-s + 3·45-s + 8·47-s + 49-s + 5·53-s − 9·55-s + 2·57-s + 13·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.458·19-s − 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s + 0.538·31-s − 0.522·33-s − 0.507·35-s + 0.986·37-s − 0.320·39-s + 0.624·41-s − 1.52·43-s + 0.447·45-s + 1.16·47-s + 1/7·49-s + 0.686·53-s − 1.21·55-s + 0.264·57-s + 1.69·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.477634777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.477634777\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−12.71331782697782, −12.03438076489790, −11.62741140839018, −10.99287609253241, −10.38336876951082, −10.20492231614422, −9.724784963053064, −9.405117115277924, −8.899856558312835, −8.435102297123779, −7.932457169905173, −7.374473739892421, −6.992519920559755, −6.448500059731789, −5.957128391901668, −5.493989465673812, −4.973088209876514, −4.655085842398512, −3.856526452698765, −3.230970050899786, −2.788238472686703, −2.256181531755715, −2.024758494915904, −1.028071283505581, −0.6337679285471626,
0.6337679285471626, 1.028071283505581, 2.024758494915904, 2.256181531755715, 2.788238472686703, 3.230970050899786, 3.856526452698765, 4.655085842398512, 4.973088209876514, 5.493989465673812, 5.957128391901668, 6.448500059731789, 6.992519920559755, 7.374473739892421, 7.932457169905173, 8.435102297123779, 8.899856558312835, 9.405117115277924, 9.724784963053064, 10.20492231614422, 10.38336876951082, 10.99287609253241, 11.62741140839018, 12.03438076489790, 12.71331782697782
