| L(s) = 1 | + 2.39·2-s + 3-s + 3.71·4-s + 3.15·5-s + 2.39·6-s + 2.42·7-s + 4.09·8-s + 9-s + 7.55·10-s − 4.78·11-s + 3.71·12-s − 2.17·13-s + 5.80·14-s + 3.15·15-s + 2.36·16-s − 2.29·17-s + 2.39·18-s + 0.938·19-s + 11.7·20-s + 2.42·21-s − 11.4·22-s − 23-s + 4.09·24-s + 4.97·25-s − 5.20·26-s + 27-s + 9.02·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.85·4-s + 1.41·5-s + 0.975·6-s + 0.918·7-s + 1.44·8-s + 0.333·9-s + 2.38·10-s − 1.44·11-s + 1.07·12-s − 0.603·13-s + 1.55·14-s + 0.815·15-s + 0.591·16-s − 0.556·17-s + 0.563·18-s + 0.215·19-s + 2.62·20-s + 0.530·21-s − 2.43·22-s − 0.208·23-s + 0.836·24-s + 0.995·25-s − 1.02·26-s + 0.192·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.179264925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.179264925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 19 | \( 1 - 0.938T + 19T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 - 8.81T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 + 9.87T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.894T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 89 | \( 1 + 5.46T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−9.264362284211904795336747649346, −8.153819013978546554297855303781, −7.46988699071205811183958117943, −6.46255771945586028120292478277, −5.74320538793597973050409425780, −4.96143892934682310712438301562, −4.59124718145859786083381808359, −3.19786255319467627779029612552, −2.40658951731279731501294274143, −1.84648719790668448469340525189,
1.84648719790668448469340525189, 2.40658951731279731501294274143, 3.19786255319467627779029612552, 4.59124718145859786083381808359, 4.96143892934682310712438301562, 5.74320538793597973050409425780, 6.46255771945586028120292478277, 7.46988699071205811183958117943, 8.153819013978546554297855303781, 9.264362284211904795336747649346
