| L(s) = 1 | + 2-s + 4-s − 4.44·7-s + 8-s − 3.44·11-s + 2.44·13-s − 4.44·14-s + 16-s + 4.44·17-s + 19-s − 3.44·22-s − 23-s + 2.44·26-s − 4.44·28-s + 4.34·29-s − 3·31-s + 32-s + 4.44·34-s + 7.79·37-s + 38-s + 0.898·41-s + 2.44·43-s − 3.44·44-s − 46-s − 7.79·47-s + 12.7·49-s + 2.44·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.68·7-s + 0.353·8-s − 1.04·11-s + 0.679·13-s − 1.18·14-s + 0.250·16-s + 1.07·17-s + 0.229·19-s − 0.735·22-s − 0.208·23-s + 0.480·26-s − 0.840·28-s + 0.807·29-s − 0.538·31-s + 0.176·32-s + 0.763·34-s + 1.28·37-s + 0.162·38-s + 0.140·41-s + 0.373·43-s − 0.520·44-s − 0.147·46-s − 1.13·47-s + 1.82·49-s + 0.339·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 - 0.898T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−7.49992796158717000267995386962, −6.37290174971030921266022870852, −6.19468822267974779959795612900, −5.43766284589915949924850337755, −4.63799093079114070206309336073, −3.71415567313684141287119058664, −3.11136007933076371377985815822, −2.63619676001836333280522118808, −1.28957214284702793809770944775, 0,
1.28957214284702793809770944775, 2.63619676001836333280522118808, 3.11136007933076371377985815822, 3.71415567313684141287119058664, 4.63799093079114070206309336073, 5.43766284589915949924850337755, 6.19468822267974779959795612900, 6.37290174971030921266022870852, 7.49992796158717000267995386962
