| L(s) = 1 | + 2.97·3-s − 3.07·5-s − 7-s + 5.83·9-s − 11-s − 13-s − 9.14·15-s + 2.23·17-s + 0.103·19-s − 2.97·21-s − 2.91·23-s + 4.47·25-s + 8.42·27-s + 5.61·29-s − 0.191·31-s − 2.97·33-s + 3.07·35-s − 6.04·37-s − 2.97·39-s − 10.4·41-s + 4.42·43-s − 17.9·45-s − 4.29·47-s + 49-s + 6.64·51-s − 0.962·53-s + 3.07·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s − 1.37·5-s − 0.377·7-s + 1.94·9-s − 0.301·11-s − 0.277·13-s − 2.36·15-s + 0.542·17-s + 0.0237·19-s − 0.648·21-s − 0.608·23-s + 0.894·25-s + 1.62·27-s + 1.04·29-s − 0.0344·31-s − 0.517·33-s + 0.520·35-s − 0.994·37-s − 0.475·39-s − 1.63·41-s + 0.675·43-s − 2.67·45-s − 0.626·47-s + 0.142·49-s + 0.930·51-s − 0.132·53-s + 0.415·55-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 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 0.103T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 + 0.191T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 0.962T + 53T^{2} \) |
| 59 | \( 1 - 7.41T + 59T^{2} \) |
| 61 | \( 1 + 6.36T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.646T + 71T^{2} \) |
| 73 | \( 1 - 2.81T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−7.55345197403233134623999837107, −7.22022814204802927348432008479, −6.35972292367315124348624545741, −5.14818720410772832414701027003, −4.35293693765477546339774568668, −3.69296829550366233626710298658, −3.19418505371999259952969212958, −2.51517401603686196991246059394, −1.43764328632949055633781768952, 0,
1.43764328632949055633781768952, 2.51517401603686196991246059394, 3.19418505371999259952969212958, 3.69296829550366233626710298658, 4.35293693765477546339774568668, 5.14818720410772832414701027003, 6.35972292367315124348624545741, 7.22022814204802927348432008479, 7.55345197403233134623999837107
