| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 7-s − 8-s + 9-s − 3·10-s − 12-s + 4·13-s − 14-s − 3·15-s + 16-s − 17-s − 18-s + 3·20-s − 21-s + 3·23-s + 24-s + 4·25-s − 4·26-s − 27-s + 28-s − 9·29-s + 3·30-s − 11·31-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 − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.670·20-s − 0.218·21-s + 0.625·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.67·29-s + 0.547·30-s − 1.97·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.99346495975361, −12.85953466589699, −11.98193597696462, −11.45678005123154, −11.00568904765707, −10.87110407360932, −10.36295080754025, −9.589790994658306, −9.494528839211788, −8.985983971178778, −8.556004577323466, −7.869202758209711, −7.375178558621993, −6.987598921799353, −6.247034756975719, −5.994701651422482, −5.590216557324019, −5.115526000175433, −4.431155206738999, −3.759116196043744, −3.223446415600260, −2.398987437032057, −1.868342604091830, −1.499066842828357, −0.8712493114171333, 0,
0.8712493114171333, 1.499066842828357, 1.868342604091830, 2.398987437032057, 3.223446415600260, 3.759116196043744, 4.431155206738999, 5.115526000175433, 5.590216557324019, 5.994701651422482, 6.247034756975719, 6.987598921799353, 7.375178558621993, 7.869202758209711, 8.556004577323466, 8.985983971178778, 9.494528839211788, 9.589790994658306, 10.36295080754025, 10.87110407360932, 11.00568904765707, 11.45678005123154, 11.98193597696462, 12.85953466589699, 12.99346495975361
