| L(s) = 1 | − 2-s − 0.372·3-s + 4-s + 3.67·5-s + 0.372·6-s − 1.58·7-s − 8-s − 2.86·9-s − 3.67·10-s + 4.52·11-s − 0.372·12-s + 0.887·13-s + 1.58·14-s − 1.36·15-s + 16-s + 4.72·17-s + 2.86·18-s + 3.99·19-s + 3.67·20-s + 0.589·21-s − 4.52·22-s − 0.0615·23-s + 0.372·24-s + 8.49·25-s − 0.887·26-s + 2.18·27-s − 1.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.215·3-s + 0.5·4-s + 1.64·5-s + 0.152·6-s − 0.597·7-s − 0.353·8-s − 0.953·9-s − 1.16·10-s + 1.36·11-s − 0.107·12-s + 0.246·13-s + 0.422·14-s − 0.353·15-s + 0.250·16-s + 1.14·17-s + 0.674·18-s + 0.915·19-s + 0.821·20-s + 0.128·21-s − 0.965·22-s − 0.0128·23-s + 0.0760·24-s + 1.69·25-s − 0.174·26-s + 0.420·27-s − 0.298·28-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
| \(L(1)\) |
\(\approx\) |
\(1.827786001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827786001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.372T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 0.887T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 + 0.0615T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 1.94T + 61T^{2} \) |
| 67 | \( 1 - 7.36T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 - 7.29T + 89T^{2} \) |
| 97 | \( 1 - 9.38T + 97T^{2} \) |
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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
−8.740746076243180608677412128322, −7.77395692048832154435100084084, −6.79699623858683602861155791571, −6.20525528151715482727581148770, −5.79672711534197376836300738049, −4.99604148885345234410368180292, −3.50291894518680325439063343384, −2.85460430855757736844007755766, −1.74305321441613622244824625832, −0.925574860996011938562403943334,
0.925574860996011938562403943334, 1.74305321441613622244824625832, 2.85460430855757736844007755766, 3.50291894518680325439063343384, 4.99604148885345234410368180292, 5.79672711534197376836300738049, 6.20525528151715482727581148770, 6.79699623858683602861155791571, 7.77395692048832154435100084084, 8.740746076243180608677412128322
