| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s − 3·9-s + 4·13-s + 2·14-s − 16-s + 3·17-s + 3·18-s − 3·19-s − 3·23-s − 4·26-s + 2·28-s − 2·29-s + 2·31-s − 5·32-s − 3·34-s + 3·36-s − 2·37-s + 3·38-s + 10·43-s + 3·46-s + 11·47-s − 3·49-s − 4·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 9-s + 1.10·13-s + 0.534·14-s − 1/4·16-s + 0.727·17-s + 0.707·18-s − 0.688·19-s − 0.625·23-s − 0.784·26-s + 0.377·28-s − 0.371·29-s + 0.359·31-s − 0.883·32-s − 0.514·34-s + 1/2·36-s − 0.328·37-s + 0.486·38-s + 1.52·43-s + 0.442·46-s + 1.60·47-s − 3/7·49-s − 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 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
−13.19676715858822, −13.00016835799957, −12.21580537815440, −12.00314021941272, −11.22767320795739, −10.80790343107781, −10.46395895366399, −9.947666148068930, −9.430713941672321, −8.944018802848899, −8.703804449525690, −8.167738775585792, −7.682482870276248, −7.255017883938043, −6.440330993815162, −6.028006943166922, −5.690611242269915, −5.052610163342031, −4.306304294435059, −3.895538265424152, −3.352593542714876, −2.767736005189749, −2.050860383496818, −1.303001414709483, −0.6522618040879662, 0,
0.6522618040879662, 1.303001414709483, 2.050860383496818, 2.767736005189749, 3.352593542714876, 3.895538265424152, 4.306304294435059, 5.052610163342031, 5.690611242269915, 6.028006943166922, 6.440330993815162, 7.255017883938043, 7.682482870276248, 8.167738775585792, 8.703804449525690, 8.944018802848899, 9.430713941672321, 9.947666148068930, 10.46395895366399, 10.80790343107781, 11.22767320795739, 12.00314021941272, 12.21580537815440, 13.00016835799957, 13.19676715858822
