| L(s) = 1 | + 2.82·3-s − 0.265·5-s − 7-s + 4.99·9-s − 11-s + 13-s − 0.749·15-s − 0.785·17-s − 3.08·19-s − 2.82·21-s − 9.36·23-s − 4.92·25-s + 5.62·27-s + 0.297·29-s + 10.2·31-s − 2.82·33-s + 0.265·35-s − 5.76·37-s + 2.82·39-s − 6.19·41-s − 12.6·43-s − 1.32·45-s − 13.3·47-s + 49-s − 2.21·51-s − 2.48·53-s + 0.265·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.118·5-s − 0.377·7-s + 1.66·9-s − 0.301·11-s + 0.277·13-s − 0.193·15-s − 0.190·17-s − 0.706·19-s − 0.616·21-s − 1.95·23-s − 0.985·25-s + 1.08·27-s + 0.0552·29-s + 1.84·31-s − 0.492·33-s + 0.0448·35-s − 0.947·37-s + 0.452·39-s − 0.967·41-s − 1.93·43-s − 0.197·45-s − 1.95·47-s + 0.142·49-s − 0.310·51-s − 0.341·53-s + 0.0357·55-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 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.265T + 5T^{2} \) |
| 17 | \( 1 + 0.785T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 + 9.36T + 23T^{2} \) |
| 29 | \( 1 - 0.297T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 + 0.731T + 89T^{2} \) |
| 97 | \( 1 + 0.817T + 97T^{2} \) |
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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
−7.952085296803102584581411688372, −6.70614598821020301797701439208, −6.46653786271501603748541575611, −5.29867715500241087688224115938, −4.38317269331141468016896609829, −3.70764199941759091549771952006, −3.17344207400602782724478120743, −2.23895908340095495325154537212, −1.69684116898056896303961352166, 0,
1.69684116898056896303961352166, 2.23895908340095495325154537212, 3.17344207400602782724478120743, 3.70764199941759091549771952006, 4.38317269331141468016896609829, 5.29867715500241087688224115938, 6.46653786271501603748541575611, 6.70614598821020301797701439208, 7.952085296803102584581411688372
