| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 7.41·5-s − 6·6-s + 8·8-s + 9·9-s − 14.8·10-s + 10.4·11-s − 12·12-s − 2.78·13-s + 22.2·15-s + 16·16-s − 50.4·17-s + 18·18-s − 125.·19-s − 29.6·20-s + 20.9·22-s − 182.·23-s − 24·24-s − 70.0·25-s − 5.57·26-s − 27·27-s + 156.·29-s + 44.4·30-s − 139.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.663·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.468·10-s + 0.287·11-s − 0.288·12-s − 0.0594·13-s + 0.382·15-s + 0.250·16-s − 0.719·17-s + 0.235·18-s − 1.50·19-s − 0.331·20-s + 0.203·22-s − 1.65·23-s − 0.204·24-s − 0.560·25-s − 0.0420·26-s − 0.192·27-s + 0.999·29-s + 0.270·30-s − 0.808·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 7.41T + 125T^{2} \) |
| 11 | \( 1 - 10.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 394.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 610.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 997.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 848.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 553.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 903.T + 9.12e5T^{2} \) |
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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
−10.98490610179967236569511807936, −10.33547488382100185905741210456, −8.881482788131080571472108435132, −7.77898897740144004195363965895, −6.70277185182823508289829043185, −5.86472909891021527139150189748, −4.54233451854330371133894481835, −3.80359878757720019177750551303, −2.04672904305431429835557206459, 0,
2.04672904305431429835557206459, 3.80359878757720019177750551303, 4.54233451854330371133894481835, 5.86472909891021527139150189748, 6.70277185182823508289829043185, 7.77898897740144004195363965895, 8.881482788131080571472108435132, 10.33547488382100185905741210456, 10.98490610179967236569511807936
