| L(s) = 1 | − 2.04·3-s − 1.32·5-s + 2.42·7-s + 1.20·9-s + 4.03·11-s + 5.65·13-s + 2.72·15-s + 17-s + 2.74·19-s − 4.97·21-s − 5.38·23-s − 3.23·25-s + 3.68·27-s − 5.83·29-s − 6.21·31-s − 8.26·33-s − 3.22·35-s − 10.4·37-s − 11.5·39-s + 7.31·41-s − 3.63·43-s − 1.59·45-s − 4.03·47-s − 1.10·49-s − 2.04·51-s − 5.87·53-s − 5.35·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s − 0.593·5-s + 0.917·7-s + 0.400·9-s + 1.21·11-s + 1.56·13-s + 0.702·15-s + 0.242·17-s + 0.629·19-s − 1.08·21-s − 1.12·23-s − 0.647·25-s + 0.709·27-s − 1.08·29-s − 1.11·31-s − 1.43·33-s − 0.545·35-s − 1.71·37-s − 1.85·39-s + 1.14·41-s − 0.555·43-s − 0.237·45-s − 0.588·47-s − 0.157·49-s − 0.287·51-s − 0.806·53-s − 0.722·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.31T + 41T^{2} \) |
| 43 | \( 1 + 3.63T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 + 5.87T + 53T^{2} \) |
| 61 | \( 1 + 1.66T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.55328920970560605110303905712, −6.59228403417974771721080552397, −6.08417284706112003146210605347, −5.47081478847600725740399298520, −4.77310639554435722917433538362, −3.79786605579174626445016409729, −3.57606922117854940036954404752, −1.81774523841302531514522908672, −1.22309532052070876262816980794, 0,
1.22309532052070876262816980794, 1.81774523841302531514522908672, 3.57606922117854940036954404752, 3.79786605579174626445016409729, 4.77310639554435722917433538362, 5.47081478847600725740399298520, 6.08417284706112003146210605347, 6.59228403417974771721080552397, 7.55328920970560605110303905712
