L(s) = 1 | + 2-s − 0.0548·3-s + 4-s + 0.621·5-s − 0.0548·6-s + 1.57·7-s + 8-s − 2.99·9-s + 0.621·10-s − 4.75·11-s − 0.0548·12-s + 4.63·13-s + 1.57·14-s − 0.0340·15-s + 16-s − 4.97·17-s − 2.99·18-s − 2.09·19-s + 0.621·20-s − 0.0861·21-s − 4.75·22-s − 1.82·23-s − 0.0548·24-s − 4.61·25-s + 4.63·26-s + 0.328·27-s + 1.57·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0316·3-s + 0.5·4-s + 0.277·5-s − 0.0223·6-s + 0.593·7-s + 0.353·8-s − 0.998·9-s + 0.196·10-s − 1.43·11-s − 0.0158·12-s + 1.28·13-s + 0.419·14-s − 0.00879·15-s + 0.250·16-s − 1.20·17-s − 0.706·18-s − 0.480·19-s + 0.138·20-s − 0.0187·21-s − 1.01·22-s − 0.379·23-s − 0.0111·24-s − 0.922·25-s + 0.909·26-s + 0.0632·27-s + 0.296·28-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 | \( 1 + 0.0548T + 3T^{2} \) |
| 5 | \( 1 - 0.621T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 - 4.63T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 3.20T + 41T^{2} \) |
| 43 | \( 1 + 7.28T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.62T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.916584369469429218171876046315, −6.68721750720519948177626156655, −6.29310998081928857335567808529, −5.45562500917260344902424316621, −4.94863401771645831473999165123, −4.14908220592806725493918020242, −3.15743928339173778941789725434, −2.47093419304656375556455301018, −1.61021325103828935206842305554, 0,
1.61021325103828935206842305554, 2.47093419304656375556455301018, 3.15743928339173778941789725434, 4.14908220592806725493918020242, 4.94863401771645831473999165123, 5.45562500917260344902424316621, 6.29310998081928857335567808529, 6.68721750720519948177626156655, 7.916584369469429218171876046315