| L(s) = 1 | + 2-s + 2.09·3-s + 4-s − 0.108·5-s + 2.09·6-s − 5.01·7-s + 8-s + 1.40·9-s − 0.108·10-s + 4.23·11-s + 2.09·12-s − 2.69·13-s − 5.01·14-s − 0.228·15-s + 16-s − 3.71·17-s + 1.40·18-s − 8.43·19-s − 0.108·20-s − 10.5·21-s + 4.23·22-s − 0.597·23-s + 2.09·24-s − 4.98·25-s − 2.69·26-s − 3.35·27-s − 5.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.21·3-s + 0.5·4-s − 0.0486·5-s + 0.856·6-s − 1.89·7-s + 0.353·8-s + 0.467·9-s − 0.0343·10-s + 1.27·11-s + 0.605·12-s − 0.746·13-s − 1.34·14-s − 0.0589·15-s + 0.250·16-s − 0.901·17-s + 0.330·18-s − 1.93·19-s − 0.0243·20-s − 2.29·21-s + 0.901·22-s − 0.124·23-s + 0.428·24-s − 0.997·25-s − 0.527·26-s − 0.645·27-s − 0.948·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 0.108T + 5T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 + 8.43T + 19T^{2} \) |
| 23 | \( 1 + 0.597T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 + 7.76T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 5.54T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.99T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 5.21T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 2.59T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.249836881583324575018995962404, −7.08659887344681644135275390197, −6.50048211888755264675029904471, −6.23467959621316132975248254341, −4.84155439349886319730989413551, −3.88099948035369610084299005244, −3.56003490355011137836238407761, −2.61757008581058181586990173031, −1.98467348337497489881432044207, 0,
1.98467348337497489881432044207, 2.61757008581058181586990173031, 3.56003490355011137836238407761, 3.88099948035369610084299005244, 4.84155439349886319730989413551, 6.23467959621316132975248254341, 6.50048211888755264675029904471, 7.08659887344681644135275390197, 8.249836881583324575018995962404
