| L(s) = 1 | − 1.00·2-s − 0.250·3-s − 0.981·4-s + 2.31·5-s + 0.252·6-s + 4.68·7-s + 3.00·8-s − 2.93·9-s − 2.33·10-s − 4.15·11-s + 0.245·12-s + 3.41·13-s − 4.72·14-s − 0.578·15-s − 1.07·16-s − 2.04·17-s + 2.96·18-s − 5.75·19-s − 2.26·20-s − 1.17·21-s + 4.19·22-s − 7.90·23-s − 0.753·24-s + 0.343·25-s − 3.44·26-s + 1.48·27-s − 4.60·28-s + ⋯ |
| L(s) = 1 | − 0.713·2-s − 0.144·3-s − 0.490·4-s + 1.03·5-s + 0.103·6-s + 1.77·7-s + 1.06·8-s − 0.979·9-s − 0.737·10-s − 1.25·11-s + 0.0709·12-s + 0.946·13-s − 1.26·14-s − 0.149·15-s − 0.267·16-s − 0.495·17-s + 0.698·18-s − 1.32·19-s − 0.507·20-s − 0.256·21-s + 0.894·22-s − 1.64·23-s − 0.153·24-s + 0.0687·25-s − 0.675·26-s + 0.285·27-s − 0.869·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2011 | \( 1 + T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 + 0.250T + 3T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + 3.72T + 37T^{2} \) |
| 41 | \( 1 + 6.39T + 41T^{2} \) |
| 43 | \( 1 - 8.36T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 0.853T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 5.62T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 + 3.50T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−8.519485044015157202151077190029, −8.309972165999790086824784171900, −7.60641267716648193147910206750, −6.21290067346744789758137502442, −5.50291859404033984055754908067, −4.90702079905396226504785468897, −3.93623429322276346707508032835, −2.21583048786529506901678200219, −1.69083614283695148031031002091, 0,
1.69083614283695148031031002091, 2.21583048786529506901678200219, 3.93623429322276346707508032835, 4.90702079905396226504785468897, 5.50291859404033984055754908067, 6.21290067346744789758137502442, 7.60641267716648193147910206750, 8.309972165999790086824784171900, 8.519485044015157202151077190029
