| L(s) = 1 | − 2.36·2-s + 3-s + 3.57·4-s + 3.36·5-s − 2.36·6-s − 7-s − 3.72·8-s + 9-s − 7.93·10-s + 5.93·11-s + 3.57·12-s + 1.42·13-s + 2.36·14-s + 3.36·15-s + 1.63·16-s + 2.78·17-s − 2.36·18-s − 5.93·19-s + 12.0·20-s − 21-s − 14.0·22-s − 23-s − 3.72·24-s + 6.29·25-s − 3.36·26-s + 27-s − 3.57·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.577·3-s + 1.78·4-s + 1.50·5-s − 0.964·6-s − 0.377·7-s − 1.31·8-s + 0.333·9-s − 2.51·10-s + 1.79·11-s + 1.03·12-s + 0.394·13-s + 0.631·14-s + 0.867·15-s + 0.409·16-s + 0.675·17-s − 0.556·18-s − 1.36·19-s + 2.68·20-s − 0.218·21-s − 2.98·22-s − 0.208·23-s − 0.759·24-s + 1.25·25-s − 0.659·26-s + 0.192·27-s − 0.675·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080044721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080044721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 9.09T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 9.15T + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 - 0.208T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.00T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 4.78T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 + 9.50T + 97T^{2} \) |
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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
−10.52666189477048796155411481167, −9.795116117014375684275452146333, −9.116418221921225878490914229367, −8.822960260355498716314835112257, −7.50686441972288628982024985433, −6.56624117934102300616147899820, −5.88928690736586041659126863633, −3.86323176938113874084946195290, −2.23578913682427248025091532263, −1.39431316014339394278105057010,
1.39431316014339394278105057010, 2.23578913682427248025091532263, 3.86323176938113874084946195290, 5.88928690736586041659126863633, 6.56624117934102300616147899820, 7.50686441972288628982024985433, 8.822960260355498716314835112257, 9.116418221921225878490914229367, 9.795116117014375684275452146333, 10.52666189477048796155411481167
