| L(s) = 1 | − 2-s − 3-s − 4-s − 3·5-s + 6-s + 7-s + 3·8-s + 9-s + 3·10-s + 11-s + 12-s − 13-s − 14-s + 3·15-s − 16-s − 18-s + 7·19-s + 3·20-s − 21-s − 22-s − 3·23-s − 3·24-s + 4·25-s + 26-s − 27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.774·15-s − 1/4·16-s − 0.235·18-s + 1.60·19-s + 0.670·20-s − 0.218·21-s − 0.213·22-s − 0.625·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.92575765230311, −13.31030030132561, −12.51305518226551, −12.28596137806202, −11.70030896946156, −11.48717486716122, −10.82515395269549, −10.37643141002065, −9.941640824743996, −9.274919468384507, −8.991107377545373, −8.167139056969330, −7.953230947565868, −7.554125859167429, −6.969866377514585, −6.483747527958650, −5.627459705254029, −5.049275231503475, −4.696557754219596, −4.171311161228521, −3.517410603225510, −3.114428639125374, −1.946418934026845, −1.298522989066728, −0.6574991615475455, 0,
0.6574991615475455, 1.298522989066728, 1.946418934026845, 3.114428639125374, 3.517410603225510, 4.171311161228521, 4.696557754219596, 5.049275231503475, 5.627459705254029, 6.483747527958650, 6.969866377514585, 7.554125859167429, 7.953230947565868, 8.167139056969330, 8.991107377545373, 9.274919468384507, 9.941640824743996, 10.37643141002065, 10.82515395269549, 11.48717486716122, 11.70030896946156, 12.28596137806202, 12.51305518226551, 13.31030030132561, 13.92575765230311
