| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 11-s − 6·13-s − 16-s − 2·17-s + 8·19-s + 2·20-s + 22-s − 25-s − 6·26-s − 29-s + 4·31-s + 5·32-s − 2·34-s − 2·37-s + 8·38-s + 6·40-s − 2·41-s − 4·43-s − 44-s − 4·47-s − 50-s + 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.66·13-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s − 1.17·26-s − 0.185·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s − 0.328·37-s + 1.29·38-s + 0.948·40-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.583·47-s − 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140679 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140679 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.385944166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385944166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.47314722923717, −12.92803722607721, −12.40456158079988, −11.94376893322884, −11.67652239809557, −11.40962336055565, −10.42743840254810, −9.970717653415413, −9.541729974574199, −9.147026342025222, −8.474470770308133, −7.982492314837971, −7.552842050888626, −6.926850233018226, −6.612913586526750, −5.686062689177698, −5.267788857527789, −4.862571480904812, −4.373823727400050, −3.695908848791204, −3.418598592242190, −2.683244194649710, −2.132628997361839, −1.018141166059722, −0.3653188903718776,
0.3653188903718776, 1.018141166059722, 2.132628997361839, 2.683244194649710, 3.418598592242190, 3.695908848791204, 4.373823727400050, 4.862571480904812, 5.267788857527789, 5.686062689177698, 6.612913586526750, 6.926850233018226, 7.552842050888626, 7.982492314837971, 8.474470770308133, 9.147026342025222, 9.541729974574199, 9.970717653415413, 10.42743840254810, 11.40962336055565, 11.67652239809557, 11.94376893322884, 12.40456158079988, 12.92803722607721, 13.47314722923717
