| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 2·11-s − 12-s + 15-s + 16-s − 17-s + 18-s + 2·19-s − 20-s − 2·22-s − 7·23-s − 24-s + 25-s − 27-s − 2·29-s + 30-s − 5·31-s + 32-s + 2·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.426·22-s − 1.45·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.898·31-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| 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
−13.09396976425024, −12.55267956505026, −12.07628741584774, −11.87062087251672, −11.37919720354278, −10.77780648492470, −10.46747265769502, −10.09767883655643, −9.356787457555732, −8.965920157842893, −8.209308420154559, −7.822257467691232, −7.391386378302317, −6.869489547132608, −6.386132929527932, −5.810085769482612, −5.396196112598006, −4.981582131989532, −4.377619787789178, −3.881539731919816, −3.420783906460368, −2.827757803132359, −2.007340353110510, −1.694370294607480, −0.6668444498422661, 0,
0.6668444498422661, 1.694370294607480, 2.007340353110510, 2.827757803132359, 3.420783906460368, 3.881539731919816, 4.377619787789178, 4.981582131989532, 5.396196112598006, 5.810085769482612, 6.386132929527932, 6.869489547132608, 7.391386378302317, 7.822257467691232, 8.209308420154559, 8.965920157842893, 9.356787457555732, 10.09767883655643, 10.46747265769502, 10.77780648492470, 11.37919720354278, 11.87062087251672, 12.07628741584774, 12.55267956505026, 13.09396976425024
