| L(s) = 1 | − 2.11·2-s − 8.38·3-s − 3.51·4-s − 10.3·5-s + 17.7·6-s + 7·7-s + 24.3·8-s + 43.2·9-s + 21.9·10-s − 48.1·11-s + 29.4·12-s − 53.1·13-s − 14.8·14-s + 87.0·15-s − 23.4·16-s − 91.6·18-s + 93.7·19-s + 36.5·20-s − 58.6·21-s + 101.·22-s − 74.3·23-s − 204.·24-s − 17.2·25-s + 112.·26-s − 136.·27-s − 24.6·28-s − 306.·29-s + ⋯ |
| L(s) = 1 | − 0.748·2-s − 1.61·3-s − 0.439·4-s − 0.928·5-s + 1.20·6-s + 0.377·7-s + 1.07·8-s + 1.60·9-s + 0.694·10-s − 1.31·11-s + 0.709·12-s − 1.13·13-s − 0.282·14-s + 1.49·15-s − 0.367·16-s − 1.20·18-s + 1.13·19-s + 0.408·20-s − 0.609·21-s + 0.987·22-s − 0.673·23-s − 1.73·24-s − 0.138·25-s + 0.848·26-s − 0.973·27-s − 0.166·28-s − 1.96·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.11T + 8T^{2} \) |
| 3 | \( 1 + 8.38T + 27T^{2} \) |
| 5 | \( 1 + 10.3T + 125T^{2} \) |
| 11 | \( 1 + 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 93.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 306.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 29.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 65.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 565.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 432.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 617.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 688.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 527.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 75.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 546.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 280.T + 9.12e5T^{2} \) |
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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
−8.036086004440395525660279685593, −7.74028016097620066819136103183, −7.13519890464525651821844614282, −5.83411374660545465240912326096, −5.09321309811867529084046934623, −4.68634183394659320236973847825, −3.60737163385825799078504844198, −1.94254848477640514308519138716, −0.60280739319928890607183630458, 0,
0.60280739319928890607183630458, 1.94254848477640514308519138716, 3.60737163385825799078504844198, 4.68634183394659320236973847825, 5.09321309811867529084046934623, 5.83411374660545465240912326096, 7.13519890464525651821844614282, 7.74028016097620066819136103183, 8.036086004440395525660279685593
