| L(s) = 1 | − 3.73·2-s − 6.19·3-s + 5.92·4-s + 20.1·5-s + 23.1·6-s + 12.7·7-s + 7.73·8-s + 11.3·9-s − 75.1·10-s + 27.1·11-s − 36.7·12-s − 73.3·13-s − 47.5·14-s − 124.·15-s − 76.2·16-s − 6.96·17-s − 42.5·18-s − 79.8·19-s + 119.·20-s − 78.8·21-s − 101.·22-s − 47.9·24-s + 279.·25-s + 273.·26-s + 96.7·27-s + 75.4·28-s + 48.2·29-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 1.19·3-s + 0.741·4-s + 1.79·5-s + 1.57·6-s + 0.687·7-s + 0.341·8-s + 0.421·9-s − 2.37·10-s + 0.743·11-s − 0.883·12-s − 1.56·13-s − 0.907·14-s − 2.14·15-s − 1.19·16-s − 0.0993·17-s − 0.556·18-s − 0.964·19-s + 1.33·20-s − 0.819·21-s − 0.981·22-s − 0.407·24-s + 2.23·25-s + 2.06·26-s + 0.689·27-s + 0.509·28-s + 0.308·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.73T + 8T^{2} \) |
| 3 | \( 1 + 6.19T + 27T^{2} \) |
| 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 - 12.7T + 343T^{2} \) |
| 11 | \( 1 - 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.96T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 48.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 69.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 32.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 485.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 950.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 765.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 652.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 405.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 590.T + 9.12e5T^{2} \) |
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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
−10.16046883289293437403870827166, −9.225746851066610406321131377292, −8.538942135627431483533937736312, −7.12248885339779831800563285350, −6.48337562813275136329375205817, −5.38869106762302284024306731161, −4.73815827400505116680843656878, −2.22650075762756716185071121546, −1.41212313178524008269997392009, 0,
1.41212313178524008269997392009, 2.22650075762756716185071121546, 4.73815827400505116680843656878, 5.38869106762302284024306731161, 6.48337562813275136329375205817, 7.12248885339779831800563285350, 8.538942135627431483533937736312, 9.225746851066610406321131377292, 10.16046883289293437403870827166
