| L(s) = 1 | + 2-s − 2.43·3-s + 4-s + 0.846·5-s − 2.43·6-s + 3.32·7-s + 8-s + 2.93·9-s + 0.846·10-s − 5.24·11-s − 2.43·12-s − 0.00893·13-s + 3.32·14-s − 2.06·15-s + 16-s + 5.00·17-s + 2.93·18-s − 1.94·19-s + 0.846·20-s − 8.09·21-s − 5.24·22-s − 8.50·23-s − 2.43·24-s − 4.28·25-s − 0.00893·26-s + 0.165·27-s + 3.32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.40·3-s + 0.5·4-s + 0.378·5-s − 0.994·6-s + 1.25·7-s + 0.353·8-s + 0.977·9-s + 0.267·10-s − 1.58·11-s − 0.703·12-s − 0.00247·13-s + 0.887·14-s − 0.532·15-s + 0.250·16-s + 1.21·17-s + 0.691·18-s − 0.446·19-s + 0.189·20-s − 1.76·21-s − 1.11·22-s − 1.77·23-s − 0.497·24-s − 0.856·25-s − 0.00175·26-s + 0.0318·27-s + 0.627·28-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 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 0.846T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 + 0.00893T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 8.50T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 + 9.47T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 0.134T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 0.680T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 0.767T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−7.79333494112899099247505575360, −7.37498263190308519185373775311, −6.15548754487977197176506573035, −5.73989387785782124196729714947, −5.15758972803603511650689383033, −4.66795055043666855882644822911, −3.62535534150063753569028208015, −2.33377193342789883424325146431, −1.51806989387651479393101582525, 0,
1.51806989387651479393101582525, 2.33377193342789883424325146431, 3.62535534150063753569028208015, 4.66795055043666855882644822911, 5.15758972803603511650689383033, 5.73989387785782124196729714947, 6.15548754487977197176506573035, 7.37498263190308519185373775311, 7.79333494112899099247505575360
