| L(s) = 1 | + 1.75·3-s − 1.19·5-s + 7-s + 0.0736·9-s + 11-s + 13-s − 2.08·15-s + 3.51·17-s − 5.94·19-s + 1.75·21-s + 1.07·23-s − 3.58·25-s − 5.13·27-s − 3.32·29-s − 2.48·31-s + 1.75·33-s − 1.19·35-s − 7.02·37-s + 1.75·39-s + 8.41·41-s − 5.42·43-s − 0.0876·45-s + 7.95·47-s + 49-s + 6.15·51-s − 12.4·53-s − 1.19·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 0.532·5-s + 0.377·7-s + 0.0245·9-s + 0.301·11-s + 0.277·13-s − 0.538·15-s + 0.851·17-s − 1.36·19-s + 0.382·21-s + 0.224·23-s − 0.716·25-s − 0.987·27-s − 0.618·29-s − 0.445·31-s + 0.305·33-s − 0.201·35-s − 1.15·37-s + 0.280·39-s + 1.31·41-s − 0.826·43-s − 0.0130·45-s + 1.16·47-s + 0.142·49-s + 0.861·51-s − 1.70·53-s − 0.160·55-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 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 4.37T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 + 5.52T + 97T^{2} \) |
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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
−7.68181700928268950506795814741, −7.01649355769564896780404989817, −6.07533716264214132033231659703, −5.44315024827475287432653847467, −4.40256705521012805957379690605, −3.81513942746056121988141886653, −3.19130957463695629846007457952, −2.26512615425100505508977732884, −1.48197457008487123710802634863, 0,
1.48197457008487123710802634863, 2.26512615425100505508977732884, 3.19130957463695629846007457952, 3.81513942746056121988141886653, 4.40256705521012805957379690605, 5.44315024827475287432653847467, 6.07533716264214132033231659703, 7.01649355769564896780404989817, 7.68181700928268950506795814741
