| L(s) = 1 | + 2.47·3-s − 0.114·5-s − 7-s + 3.11·9-s − 11-s + 13-s − 0.284·15-s − 5.83·17-s + 2.75·19-s − 2.47·21-s + 3.81·23-s − 4.98·25-s + 0.284·27-s − 5.17·29-s − 5.35·31-s − 2.47·33-s + 0.114·35-s + 2.64·37-s + 2.47·39-s − 8.79·41-s + 9.34·43-s − 0.357·45-s − 5.51·47-s + 49-s − 14.4·51-s − 13.0·53-s + 0.114·55-s + ⋯ |
| L(s) = 1 | + 1.42·3-s − 0.0513·5-s − 0.377·7-s + 1.03·9-s − 0.301·11-s + 0.277·13-s − 0.0733·15-s − 1.41·17-s + 0.632·19-s − 0.539·21-s + 0.796·23-s − 0.997·25-s + 0.0546·27-s − 0.961·29-s − 0.962·31-s − 0.430·33-s + 0.0194·35-s + 0.434·37-s + 0.395·39-s − 1.37·41-s + 1.42·43-s − 0.0533·45-s − 0.804·47-s + 0.142·49-s − 2.01·51-s − 1.78·53-s + 0.0154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 5 | \( 1 + 0.114T + 5T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 2.75T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 + 8.79T + 41T^{2} \) |
| 43 | \( 1 - 9.34T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 8.64T + 79T^{2} \) |
| 83 | \( 1 - 9.93T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−7.60357351096901893010620028373, −7.02010479557464569627584213668, −6.20159089444985776303165611494, −5.36866081337490691864172793580, −4.44990077703980957017741540924, −3.69750902532459599530340402237, −3.12192397708819624298116069125, −2.32901207350468195015805086736, −1.59485021073848719015654026179, 0,
1.59485021073848719015654026179, 2.32901207350468195015805086736, 3.12192397708819624298116069125, 3.69750902532459599530340402237, 4.44990077703980957017741540924, 5.36866081337490691864172793580, 6.20159089444985776303165611494, 7.02010479557464569627584213668, 7.60357351096901893010620028373
