| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 2·13-s − 15-s + 16-s − 2·17-s + 18-s + 19-s + 20-s − 4·22-s − 24-s + 25-s + 2·26-s − 27-s + 6·29-s − 30-s + 4·31-s + 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.99649214683175, −12.56080330910013, −12.11199019953257, −11.41849599370906, −11.13751051344301, −10.82119474340994, −10.19337339188005, −9.900820060199529, −9.353242871135786, −8.735372933635644, −8.154899786706738, −7.698624204553013, −7.364305109768503, −6.461903822270885, −6.247060711365598, −6.021966416672445, −5.156033168711019, −4.913377476776621, −4.509148715994160, −3.859765875949874, −3.124562861104052, −2.730094424781593, −2.195977677148509, −1.441707829064450, −0.8675080387887004, 0,
0.8675080387887004, 1.441707829064450, 2.195977677148509, 2.730094424781593, 3.124562861104052, 3.859765875949874, 4.509148715994160, 4.913377476776621, 5.156033168711019, 6.021966416672445, 6.247060711365598, 6.461903822270885, 7.364305109768503, 7.698624204553013, 8.154899786706738, 8.735372933635644, 9.353242871135786, 9.900820060199529, 10.19337339188005, 10.82119474340994, 11.13751051344301, 11.41849599370906, 12.11199019953257, 12.56080330910013, 12.99649214683175
