| L(s) = 1 | + 2-s − 3.27·3-s + 4-s + 0.938·5-s − 3.27·6-s + 3.32·7-s + 8-s + 7.71·9-s + 0.938·10-s − 3.16·11-s − 3.27·12-s − 6.32·13-s + 3.32·14-s − 3.07·15-s + 16-s + 2.28·17-s + 7.71·18-s + 6.76·19-s + 0.938·20-s − 10.8·21-s − 3.16·22-s + 1.03·23-s − 3.27·24-s − 4.11·25-s − 6.32·26-s − 15.4·27-s + 3.32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.89·3-s + 0.5·4-s + 0.419·5-s − 1.33·6-s + 1.25·7-s + 0.353·8-s + 2.57·9-s + 0.296·10-s − 0.954·11-s − 0.945·12-s − 1.75·13-s + 0.888·14-s − 0.793·15-s + 0.250·16-s + 0.553·17-s + 1.81·18-s + 1.55·19-s + 0.209·20-s − 2.37·21-s − 0.674·22-s + 0.215·23-s − 0.668·24-s − 0.823·25-s − 1.24·26-s − 2.97·27-s + 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 0.938T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 - 1.03T + 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 0.0865T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 0.781T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 0.322T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 - 9.55T + 79T^{2} \) |
| 83 | \( 1 - 2.38T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 3.92T + 97T^{2} \) |
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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
−7.60266366713660554755286702756, −6.92789271881302733771537092750, −5.96337686852865059275837248429, −5.40346798113635700590176830545, −4.92075522330956480744642759275, −4.70985760324147973812811532623, −3.37353774035921844144822567087, −2.11562947487463823586273665212, −1.35772884548552472839810597460, 0,
1.35772884548552472839810597460, 2.11562947487463823586273665212, 3.37353774035921844144822567087, 4.70985760324147973812811532623, 4.92075522330956480744642759275, 5.40346798113635700590176830545, 5.96337686852865059275837248429, 6.92789271881302733771537092750, 7.60266366713660554755286702756
