L(s) = 1 | − 2.58·2-s + 3-s + 4.67·4-s + 2.37·5-s − 2.58·6-s − 1.51·7-s − 6.90·8-s + 9-s − 6.12·10-s + 0.0286·11-s + 4.67·12-s − 2.28·13-s + 3.90·14-s + 2.37·15-s + 8.47·16-s − 17-s − 2.58·18-s + 3.73·19-s + 11.0·20-s − 1.51·21-s − 0.0740·22-s + 0.0338·23-s − 6.90·24-s + 0.629·25-s + 5.89·26-s + 27-s − 7.06·28-s + ⋯ |
L(s) = 1 | − 1.82·2-s + 0.577·3-s + 2.33·4-s + 1.06·5-s − 1.05·6-s − 0.571·7-s − 2.43·8-s + 0.333·9-s − 1.93·10-s + 0.00864·11-s + 1.34·12-s − 0.632·13-s + 1.04·14-s + 0.612·15-s + 2.11·16-s − 0.242·17-s − 0.608·18-s + 0.857·19-s + 2.47·20-s − 0.330·21-s − 0.0157·22-s + 0.00705·23-s − 1.40·24-s + 0.125·25-s + 1.15·26-s + 0.192·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 0.0286T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 0.0338T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 9.63T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 - 7.88T + 79T^{2} \) |
| 83 | \( 1 + 6.05T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−7.67944404950283513131728065451, −7.02145015289742420863419001714, −6.40985286718794585102397585822, −5.77008192548688288504581293794, −4.77922210634458637516550149206, −3.41792616901272457118068825100, −2.70861467015625811457556328967, −2.00174487909884486182711250773, −1.27874478019351404392992755590, 0,
1.27874478019351404392992755590, 2.00174487909884486182711250773, 2.70861467015625811457556328967, 3.41792616901272457118068825100, 4.77922210634458637516550149206, 5.77008192548688288504581293794, 6.40985286718794585102397585822, 7.02145015289742420863419001714, 7.67944404950283513131728065451