| L(s) = 1 | − 0.618·2-s − 9.18·3-s − 7.61·4-s + 19.4·5-s + 5.67·6-s + 15.1·7-s + 9.65·8-s + 57.2·9-s − 12.0·10-s − 7.19·11-s + 69.9·12-s + 36.4·13-s − 9.37·14-s − 178.·15-s + 54.9·16-s + 102.·17-s − 35.4·18-s − 51.6·19-s − 147.·20-s − 139.·21-s + 4.44·22-s − 102.·23-s − 88.6·24-s + 252.·25-s − 22.5·26-s − 277.·27-s − 115.·28-s + ⋯ |
| L(s) = 1 | − 0.218·2-s − 1.76·3-s − 0.952·4-s + 1.73·5-s + 0.386·6-s + 0.818·7-s + 0.426·8-s + 2.12·9-s − 0.379·10-s − 0.197·11-s + 1.68·12-s + 0.777·13-s − 0.178·14-s − 3.06·15-s + 0.858·16-s + 1.45·17-s − 0.463·18-s − 0.624·19-s − 1.65·20-s − 1.44·21-s + 0.0431·22-s − 0.929·23-s − 0.753·24-s + 2.01·25-s − 0.170·26-s − 1.98·27-s − 0.779·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 8T^{2} \) |
| 3 | \( 1 + 9.18T + 27T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 15.1T + 343T^{2} \) |
| 11 | \( 1 + 7.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 39.6T + 6.89e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 124.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 838.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 215.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 494.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 712.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 431.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−8.589129943717091093411669528754, −7.71040766087249027722448633037, −6.54338214348394639875568169935, −5.86342057853144007979042290810, −5.32766904040861112922327687501, −4.87455836036078955995958336530, −3.70667130744666053619056843778, −1.73940532584875350653739750030, −1.26550482555400419440134412325, 0,
1.26550482555400419440134412325, 1.73940532584875350653739750030, 3.70667130744666053619056843778, 4.87455836036078955995958336530, 5.32766904040861112922327687501, 5.86342057853144007979042290810, 6.54338214348394639875568169935, 7.71040766087249027722448633037, 8.589129943717091093411669528754
