| L(s) = 1 | − 2.31·2-s + 3.36·4-s + 2.07·5-s + 1.30·7-s − 3.16·8-s − 4.81·10-s + 0.169·11-s + 0.705·13-s − 3.02·14-s + 0.600·16-s + 2.81·17-s − 7.73·19-s + 6.99·20-s − 0.392·22-s − 23-s − 0.686·25-s − 1.63·26-s + 4.38·28-s − 29-s − 4.11·31-s + 4.94·32-s − 6.51·34-s + 2.70·35-s + 4.93·37-s + 17.9·38-s − 6.57·40-s + 2.38·41-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.68·4-s + 0.928·5-s + 0.492·7-s − 1.11·8-s − 1.52·10-s + 0.0511·11-s + 0.195·13-s − 0.807·14-s + 0.150·16-s + 0.682·17-s − 1.77·19-s + 1.56·20-s − 0.0837·22-s − 0.208·23-s − 0.137·25-s − 0.320·26-s + 0.829·28-s − 0.185·29-s − 0.738·31-s + 0.873·32-s − 1.11·34-s + 0.457·35-s + 0.811·37-s + 2.90·38-s − 1.03·40-s + 0.372·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 0.169T + 11T^{2} \) |
| 13 | \( 1 - 0.705T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 31 | \( 1 + 4.11T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 8.28T + 97T^{2} \) |
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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
−7.87309876934518923950858586044, −7.33100549367681991843156931621, −6.29207543507345333119023811340, −6.03961905172673049364849106642, −4.93399983524538499553942747241, −4.01834189088853907211771921330, −2.72291563263625661682550116914, −1.92590745049594204379979453160, −1.34631087215494428777951036907, 0,
1.34631087215494428777951036907, 1.92590745049594204379979453160, 2.72291563263625661682550116914, 4.01834189088853907211771921330, 4.93399983524538499553942747241, 6.03961905172673049364849106642, 6.29207543507345333119023811340, 7.33100549367681991843156931621, 7.87309876934518923950858586044
