| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2.24·11-s + 5.65·13-s + 16-s − 2.58·17-s − 6.82·19-s − 20-s − 2.24·22-s − 3.17·23-s + 25-s + 5.65·26-s − 2.58·29-s − 10.2·31-s + 32-s − 2.58·34-s − 0.242·37-s − 6.82·38-s − 40-s − 10.4·41-s + 9.07·43-s − 2.24·44-s − 3.17·46-s + 2.24·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.676·11-s + 1.56·13-s + 0.250·16-s − 0.627·17-s − 1.56·19-s − 0.223·20-s − 0.478·22-s − 0.661·23-s + 0.200·25-s + 1.10·26-s − 0.480·29-s − 1.83·31-s + 0.176·32-s − 0.443·34-s − 0.0398·37-s − 1.10·38-s − 0.158·40-s − 1.63·41-s + 1.38·43-s − 0.338·44-s − 0.467·46-s + 0.327·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.07T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−8.014271634716101688643466501802, −7.16110335563032482624524298645, −6.41173442360323623080883433038, −5.80558055176035274037706594522, −4.99486987651787635951778614392, −3.99168586714970919692232342152, −3.70476241105780270121908962220, −2.51061622990782702433032969486, −1.63575319941075512526310860312, 0,
1.63575319941075512526310860312, 2.51061622990782702433032969486, 3.70476241105780270121908962220, 3.99168586714970919692232342152, 4.99486987651787635951778614392, 5.80558055176035274037706594522, 6.41173442360323623080883433038, 7.16110335563032482624524298645, 8.014271634716101688643466501802
