| L(s) = 1 | + 2-s − 2.41·3-s + 4-s − 2.86·5-s − 2.41·6-s + 3.82·7-s + 8-s + 2.85·9-s − 2.86·10-s + 0.871·11-s − 2.41·12-s + 2.45·13-s + 3.82·14-s + 6.93·15-s + 16-s + 6.32·17-s + 2.85·18-s − 4.50·19-s − 2.86·20-s − 9.25·21-s + 0.871·22-s + 2.90·23-s − 2.41·24-s + 3.22·25-s + 2.45·26-s + 0.355·27-s + 3.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 1.28·5-s − 0.987·6-s + 1.44·7-s + 0.353·8-s + 0.951·9-s − 0.906·10-s + 0.262·11-s − 0.698·12-s + 0.679·13-s + 1.02·14-s + 1.79·15-s + 0.250·16-s + 1.53·17-s + 0.672·18-s − 1.03·19-s − 0.641·20-s − 2.01·21-s + 0.185·22-s + 0.605·23-s − 0.493·24-s + 0.644·25-s + 0.480·26-s + 0.0683·27-s + 0.723·28-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
| \(L(1)\) |
\(\approx\) |
\(1.746106159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746106159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 0.871T + 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 + 9.51T + 29T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 + 0.968T + 37T^{2} \) |
| 41 | \( 1 - 0.708T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 + 9.98T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 + 2.04T + 73T^{2} \) |
| 79 | \( 1 - 8.86T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 - 5.71T + 89T^{2} \) |
| 97 | \( 1 - 9.08T + 97T^{2} \) |
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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
−8.141722660378535656595993643328, −7.63246106027670246220312825871, −6.92147795564258842548306362803, −5.98310766943813062220559843390, −5.42565236541169803287357044599, −4.70469181338739843535057412018, −4.11212087106960977049665627293, −3.33430663149530539369024717837, −1.75946973472311407167624037514, −0.76784991018486363863742699044,
0.76784991018486363863742699044, 1.75946973472311407167624037514, 3.33430663149530539369024717837, 4.11212087106960977049665627293, 4.70469181338739843535057412018, 5.42565236541169803287357044599, 5.98310766943813062220559843390, 6.92147795564258842548306362803, 7.63246106027670246220312825871, 8.141722660378535656595993643328
