| L(s) = 1 | + 2-s + 1.76·3-s + 4-s − 2.65·5-s + 1.76·6-s − 0.218·7-s + 8-s + 0.0989·9-s − 2.65·10-s + 4.66·11-s + 1.76·12-s − 3.93·13-s − 0.218·14-s − 4.67·15-s + 16-s − 2.41·17-s + 0.0989·18-s − 6.80·19-s − 2.65·20-s − 0.385·21-s + 4.66·22-s − 5.25·23-s + 1.76·24-s + 2.05·25-s − 3.93·26-s − 5.10·27-s − 0.218·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.01·3-s + 0.5·4-s − 1.18·5-s + 0.718·6-s − 0.0826·7-s + 0.353·8-s + 0.0329·9-s − 0.840·10-s + 1.40·11-s + 0.508·12-s − 1.09·13-s − 0.0584·14-s − 1.20·15-s + 0.250·16-s − 0.585·17-s + 0.0233·18-s − 1.56·19-s − 0.594·20-s − 0.0840·21-s + 0.994·22-s − 1.09·23-s + 0.359·24-s + 0.411·25-s − 0.771·26-s − 0.982·27-s − 0.0413·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 + 0.218T + 7T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 + 0.320T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 + 6.22T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 + 0.570T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 0.793T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 - 4.52T + 89T^{2} \) |
| 97 | \( 1 - 3.73T + 97T^{2} \) |
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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
−8.162106656780519071854152556097, −7.35489817063757780182182223325, −6.68407840870252340201397860334, −5.96503388490047971011316071566, −4.65272740869058146140514264614, −4.14141349483904005561617847002, −3.58672133318093776482604342406, −2.64926641593597708521970069765, −1.84067520215919286789303489433, 0,
1.84067520215919286789303489433, 2.64926641593597708521970069765, 3.58672133318093776482604342406, 4.14141349483904005561617847002, 4.65272740869058146140514264614, 5.96503388490047971011316071566, 6.68407840870252340201397860334, 7.35489817063757780182182223325, 8.162106656780519071854152556097
