| L(s) = 1 | + 2.88·3-s − 0.992·5-s − 7-s + 5.33·9-s − 3.42·11-s − 2.77·13-s − 2.86·15-s + 6.93·17-s − 1.96·19-s − 2.88·21-s + 2.05·23-s − 4.01·25-s + 6.73·27-s − 7.55·29-s + 5.23·31-s − 9.87·33-s + 0.992·35-s − 9.31·37-s − 8.00·39-s + 0.949·41-s − 8.41·43-s − 5.29·45-s − 4.64·47-s + 49-s + 20.0·51-s − 10.2·53-s + 3.39·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.443·5-s − 0.377·7-s + 1.77·9-s − 1.03·11-s − 0.769·13-s − 0.739·15-s + 1.68·17-s − 0.450·19-s − 0.629·21-s + 0.428·23-s − 0.803·25-s + 1.29·27-s − 1.40·29-s + 0.940·31-s − 1.71·33-s + 0.167·35-s − 1.53·37-s − 1.28·39-s + 0.148·41-s − 1.28·43-s − 0.789·45-s − 0.677·47-s + 0.142·49-s + 2.80·51-s − 1.40·53-s + 0.457·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 + 0.992T + 5T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 - 2.05T + 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 9.31T + 37T^{2} \) |
| 41 | \( 1 - 0.949T + 41T^{2} \) |
| 43 | \( 1 + 8.41T + 43T^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 6.35T + 83T^{2} \) |
| 89 | \( 1 + 0.428T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.66811795494374039713061861714, −7.33828081627247206713787212423, −6.31177161504815995918464233697, −5.29556383673660672909427428085, −4.63438003747961825608851366420, −3.43785933999198257971508918995, −3.37497448620879548930416280935, −2.41732813375273055149432273051, −1.61103958435641430702985039821, 0,
1.61103958435641430702985039821, 2.41732813375273055149432273051, 3.37497448620879548930416280935, 3.43785933999198257971508918995, 4.63438003747961825608851366420, 5.29556383673660672909427428085, 6.31177161504815995918464233697, 7.33828081627247206713787212423, 7.66811795494374039713061861714
