| L(s) = 1 | − 1.31·2-s + 3·3-s − 6.28·4-s + 5.66·5-s − 3.93·6-s − 0.126·7-s + 18.7·8-s + 9·9-s − 7.43·10-s + 11·11-s − 18.8·12-s − 16.0·13-s + 0.165·14-s + 17.0·15-s + 25.7·16-s − 70.6·17-s − 11.7·18-s + 53.4·19-s − 35.6·20-s − 0.378·21-s − 14.4·22-s − 26.8·23-s + 56.1·24-s − 92.8·25-s + 21.0·26-s + 27·27-s + 0.792·28-s + ⋯ |
| L(s) = 1 | − 0.463·2-s + 0.577·3-s − 0.785·4-s + 0.506·5-s − 0.267·6-s − 0.00680·7-s + 0.827·8-s + 0.333·9-s − 0.234·10-s + 0.301·11-s − 0.453·12-s − 0.343·13-s + 0.00315·14-s + 0.292·15-s + 0.401·16-s − 1.00·17-s − 0.154·18-s + 0.645·19-s − 0.398·20-s − 0.00393·21-s − 0.139·22-s − 0.243·23-s + 0.477·24-s − 0.743·25-s + 0.159·26-s + 0.192·27-s + 0.00534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 1.31T + 8T^{2} \) |
| 5 | \( 1 - 5.66T + 125T^{2} \) |
| 7 | \( 1 + 0.126T + 343T^{2} \) |
| 13 | \( 1 + 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 53.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 13.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 97.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 681.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 51.1T + 2.05e5T^{2} \) |
| 67 | \( 1 - 306.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 860.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 45.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 16.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 442.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 155.T + 9.12e5T^{2} \) |
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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
−8.521623890846261356260040961854, −7.84325532043378073097272392055, −7.00785666744542092492363826137, −6.07347270561599317151245830147, −5.02885047571559353785358904625, −4.33700989007615570440625484404, −3.37345177783502078742295469789, −2.22268386861745403534304324146, −1.26290687635751771787073455597, 0,
1.26290687635751771787073455597, 2.22268386861745403534304324146, 3.37345177783502078742295469789, 4.33700989007615570440625484404, 5.02885047571559353785358904625, 6.07347270561599317151245830147, 7.00785666744542092492363826137, 7.84325532043378073097272392055, 8.521623890846261356260040961854
