L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 4·7-s − 3·8-s − 2·9-s + 10-s − 2·11-s − 12-s + 2·13-s + 4·14-s + 15-s − 16-s + 3·17-s − 2·18-s + 5·19-s − 20-s + 4·21-s − 2·22-s + 7·23-s − 3·24-s − 4·25-s + 2·26-s − 5·27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.872·21-s − 0.426·22-s + 1.45·23-s − 0.612·24-s − 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.45636762139405, −13.28576674964887, −12.71600218860068, −12.08171556214329, −11.61568642850704, −11.14667274016407, −10.89244570336737, −10.07578430689845, −9.456356265847952, −9.251378773433749, −8.652540804859988, −8.087621311896879, −7.913718071762976, −7.324972988418656, −6.545721715530421, −5.832550691347391, −5.378429988412966, −5.177173007087387, −4.721272054891493, −3.809978751521974, −3.511773266405766, −2.909133076067920, −2.334162845756546, −1.584441878088868, −1.034828437743972, 0,
1.034828437743972, 1.584441878088868, 2.334162845756546, 2.909133076067920, 3.511773266405766, 3.809978751521974, 4.721272054891493, 5.177173007087387, 5.378429988412966, 5.832550691347391, 6.545721715530421, 7.324972988418656, 7.913718071762976, 8.087621311896879, 8.652540804859988, 9.251378773433749, 9.456356265847952, 10.07578430689845, 10.89244570336737, 11.14667274016407, 11.61568642850704, 12.08171556214329, 12.71600218860068, 13.28576674964887, 13.45636762139405