| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·7-s − 3·8-s + 9-s − 2·11-s − 12-s − 13-s − 3·14-s − 16-s + 18-s + 5·19-s − 3·21-s − 2·22-s + 3·23-s − 3·24-s − 26-s + 27-s + 3·28-s + 4·29-s + 4·31-s + 5·32-s − 2·33-s − 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 1/4·16-s + 0.235·18-s + 1.14·19-s − 0.654·21-s − 0.426·22-s + 0.625·23-s − 0.612·24-s − 0.196·26-s + 0.192·27-s + 0.566·28-s + 0.742·29-s + 0.718·31-s + 0.883·32-s − 0.348·33-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.95792572167512, −12.70578604633754, −12.15358113777083, −11.93944059708002, −11.14244629226096, −10.61910391744576, −10.10565986148437, −9.619233736841630, −9.334644574294131, −8.973820712097830, −8.333220913766268, −7.768564047587290, −7.503439982852128, −6.636905620584262, −6.449120995372212, −5.780897862250137, −5.299671464345294, −4.780000434585277, −4.362247349285538, −3.688265611532973, −3.261250898848967, −2.723891817023044, −2.585934952536099, −1.406722244384717, −0.7435772058695952, 0,
0.7435772058695952, 1.406722244384717, 2.585934952536099, 2.723891817023044, 3.261250898848967, 3.688265611532973, 4.362247349285538, 4.780000434585277, 5.299671464345294, 5.780897862250137, 6.449120995372212, 6.636905620584262, 7.503439982852128, 7.768564047587290, 8.333220913766268, 8.973820712097830, 9.334644574294131, 9.619233736841630, 10.10565986148437, 10.61910391744576, 11.14244629226096, 11.93944059708002, 12.15358113777083, 12.70578604633754, 12.95792572167512
