| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s + 3·11-s + 12-s − 4·13-s + 14-s − 2·15-s + 16-s + 18-s − 8·19-s − 2·20-s + 21-s + 3·22-s − 2·23-s + 24-s − 25-s − 4·26-s + 27-s + 28-s − 9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s − 0.447·20-s + 0.218·21-s + 0.639·22-s − 0.417·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.188·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 31 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.77739559920817962199524728678, −7.04963054930073881618287304813, −6.44649567486306005787749415542, −5.49238901980173339180666664307, −4.56406115225960097349459258412, −4.07698481292536347614536523243, −3.49202302391071279338294181643, −2.38133531738909960778620705471, −1.70472214859375948819832239929, 0,
1.70472214859375948819832239929, 2.38133531738909960778620705471, 3.49202302391071279338294181643, 4.07698481292536347614536523243, 4.56406115225960097349459258412, 5.49238901980173339180666664307, 6.44649567486306005787749415542, 7.04963054930073881618287304813, 7.77739559920817962199524728678
