| L(s) = 1 | − 2-s + 1.96·3-s + 4-s + 1.63·5-s − 1.96·6-s − 8-s + 0.871·9-s − 1.63·10-s + 0.177·11-s + 1.96·12-s − 5.66·13-s + 3.22·15-s + 16-s − 6.01·17-s − 0.871·18-s − 5.54·19-s + 1.63·20-s − 0.177·22-s + 6.91·23-s − 1.96·24-s − 2.31·25-s + 5.66·26-s − 4.18·27-s + 6.25·29-s − 3.22·30-s + 2.73·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s + 0.732·5-s − 0.803·6-s − 0.353·8-s + 0.290·9-s − 0.517·10-s + 0.0534·11-s + 0.568·12-s − 1.57·13-s + 0.832·15-s + 0.250·16-s − 1.45·17-s − 0.205·18-s − 1.27·19-s + 0.366·20-s − 0.0377·22-s + 1.44·23-s − 0.401·24-s − 0.463·25-s + 1.11·26-s − 0.805·27-s + 1.16·29-s − 0.588·30-s + 0.491·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 - 0.177T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 6.01T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 43 | \( 1 - 0.629T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 0.178T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.408200860579734169892327166237, −7.36454767316920870576645571970, −6.89908879323825387708951000835, −6.07252296960436407303630013899, −5.01610957084730065492392827597, −4.22639047664086294864741751244, −2.91791108296469401859190988673, −2.44919984539721263686371725163, −1.71827832647093179691022993915, 0,
1.71827832647093179691022993915, 2.44919984539721263686371725163, 2.91791108296469401859190988673, 4.22639047664086294864741751244, 5.01610957084730065492392827597, 6.07252296960436407303630013899, 6.89908879323825387708951000835, 7.36454767316920870576645571970, 8.408200860579734169892327166237
