| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 3·14-s + 15-s + 16-s + 17-s + 18-s − 20-s − 3·21-s + 22-s + 23-s − 24-s − 4·25-s + 26-s − 27-s + 3·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.223·20-s − 0.654·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.38502616283872, −15.61446840170493, −15.26383146207616, −14.76306351871739, −14.11208242639447, −13.64872141580504, −12.98374580032091, −12.37898075304627, −11.70730164554611, −11.48080122020032, −11.01178803369538, −10.23734936821670, −9.731532444800512, −8.642215179882448, −8.301497490866289, −7.449070521042309, −7.073896203546608, −6.237606437930419, −5.602304674154404, −5.023259066288295, −4.451345296882131, −3.777182290825180, −3.098026536880101, −1.857813460979764, −1.407922907920762, 0,
1.407922907920762, 1.857813460979764, 3.098026536880101, 3.777182290825180, 4.451345296882131, 5.023259066288295, 5.602304674154404, 6.237606437930419, 7.073896203546608, 7.449070521042309, 8.301497490866289, 8.642215179882448, 9.731532444800512, 10.23734936821670, 11.01178803369538, 11.48080122020032, 11.70730164554611, 12.37898075304627, 12.98374580032091, 13.64872141580504, 14.11208242639447, 14.76306351871739, 15.26383146207616, 15.61446840170493, 16.38502616283872
