L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s − 12-s − 13-s − 4·14-s + 15-s + 16-s − 6·17-s + 18-s − 4·19-s − 20-s + 4·21-s − 4·23-s − 24-s + 25-s − 26-s − 27-s − 4·28-s + 2·29-s + 30-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 + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.97416146655951, −14.26845982457994, −13.62831574919108, −13.26478048321491, −12.79903103002207, −12.24348694779029, −11.99205783451513, −11.28900765745388, −10.78306564377545, −10.25324197457856, −9.805660106008881, −9.145275425635992, −8.510521466968704, −7.907402762402643, −7.107464157849696, −6.616850497858247, −6.369862621497195, −5.804161949856204, −5.015428578865492, −4.304786539021810, −4.080226600977131, −3.252192775276564, −2.596773289685143, −2.016478710479035, −0.7433698810950821, 0,
0.7433698810950821, 2.016478710479035, 2.596773289685143, 3.252192775276564, 4.080226600977131, 4.304786539021810, 5.015428578865492, 5.804161949856204, 6.369862621497195, 6.616850497858247, 7.107464157849696, 7.907402762402643, 8.510521466968704, 9.145275425635992, 9.805660106008881, 10.25324197457856, 10.78306564377545, 11.28900765745388, 11.99205783451513, 12.24348694779029, 12.79903103002207, 13.26478048321491, 13.62831574919108, 14.26845982457994, 14.97416146655951