| L(s) = 1 | − 3-s + 1.94·5-s − 4.16·7-s + 9-s − 3.61·11-s + 2.66·13-s − 1.94·15-s − 0.377·17-s + 0.763·19-s + 4.16·21-s + 5.16·23-s − 1.21·25-s − 27-s + 3.19·29-s + 8.45·31-s + 3.61·33-s − 8.10·35-s + 3.83·37-s − 2.66·39-s − 7.23·41-s − 6.75·43-s + 1.94·45-s − 4.50·47-s + 10.3·49-s + 0.377·51-s + 11.0·53-s − 7.03·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.870·5-s − 1.57·7-s + 0.333·9-s − 1.09·11-s + 0.739·13-s − 0.502·15-s − 0.0915·17-s + 0.175·19-s + 0.908·21-s + 1.07·23-s − 0.243·25-s − 0.192·27-s + 0.592·29-s + 1.51·31-s + 0.629·33-s − 1.36·35-s + 0.630·37-s − 0.427·39-s − 1.13·41-s − 1.03·43-s + 0.290·45-s − 0.657·47-s + 1.47·49-s + 0.0528·51-s + 1.51·53-s − 0.948·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.94T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 + 0.377T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 5.16T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 - 8.45T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 + 5.97T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−8.145590024207523363453787582691, −7.04799825001877961386181550623, −6.52811894335002170912768116906, −5.90599063967130064203959310806, −5.30575581473422156863942942533, −4.36636322797964075946104754939, −3.19263689206877121010231434894, −2.65883176599612271944190032975, −1.29072487708857856261308499777, 0,
1.29072487708857856261308499777, 2.65883176599612271944190032975, 3.19263689206877121010231434894, 4.36636322797964075946104754939, 5.30575581473422156863942942533, 5.90599063967130064203959310806, 6.52811894335002170912768116906, 7.04799825001877961386181550623, 8.145590024207523363453787582691
