| L(s) = 1 | − 2-s + 3.09·3-s + 4-s − 1.23·5-s − 3.09·6-s − 4.11·7-s − 8-s + 6.60·9-s + 1.23·10-s − 2.95·11-s + 3.09·12-s − 4.28·13-s + 4.11·14-s − 3.82·15-s + 16-s + 7.58·17-s − 6.60·18-s + 6.52·19-s − 1.23·20-s − 12.7·21-s + 2.95·22-s + 3.37·23-s − 3.09·24-s − 3.47·25-s + 4.28·26-s + 11.1·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·3-s + 0.5·4-s − 0.552·5-s − 1.26·6-s − 1.55·7-s − 0.353·8-s + 2.20·9-s + 0.390·10-s − 0.890·11-s + 0.894·12-s − 1.18·13-s + 1.09·14-s − 0.987·15-s + 0.250·16-s + 1.84·17-s − 1.55·18-s + 1.49·19-s − 0.276·20-s − 2.78·21-s + 0.629·22-s + 0.702·23-s − 0.632·24-s − 0.695·25-s + 0.840·26-s + 2.14·27-s − 0.777·28-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 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 - 0.398T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 + 5.73T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 1.59T + 97T^{2} \) |
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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
−7.82813032212501279741029257115, −7.54444157955268183910642665145, −7.30775041515244213675343171847, −6.02209631558834966992073410622, −5.03902942039716922994159496943, −3.71125260485163949739548796950, −3.10833889444662082849171909861, −2.82201827502530372225194348870, −1.53643246181142282152936166983, 0,
1.53643246181142282152936166983, 2.82201827502530372225194348870, 3.10833889444662082849171909861, 3.71125260485163949739548796950, 5.03902942039716922994159496943, 6.02209631558834966992073410622, 7.30775041515244213675343171847, 7.54444157955268183910642665145, 7.82813032212501279741029257115
