| L(s) = 1 | + 2.14·2-s − 3-s + 2.60·4-s − 3.79·5-s − 2.14·6-s − 4.31·7-s + 1.30·8-s + 9-s − 8.14·10-s − 4.29·11-s − 2.60·12-s + 1.10·13-s − 9.27·14-s + 3.79·15-s − 2.41·16-s + 17-s + 2.14·18-s + 4.02·19-s − 9.89·20-s + 4.31·21-s − 9.21·22-s + 7.86·23-s − 1.30·24-s + 9.38·25-s + 2.37·26-s − 27-s − 11.2·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.30·4-s − 1.69·5-s − 0.876·6-s − 1.63·7-s + 0.461·8-s + 0.333·9-s − 2.57·10-s − 1.29·11-s − 0.753·12-s + 0.306·13-s − 2.47·14-s + 0.979·15-s − 0.603·16-s + 0.242·17-s + 0.505·18-s + 0.922·19-s − 2.21·20-s + 0.942·21-s − 1.96·22-s + 1.63·23-s − 0.266·24-s + 1.87·25-s + 0.464·26-s − 0.192·27-s − 2.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315962187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315962187\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 - 4.79T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 - 9.81T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + 0.101T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 0.207T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 - 2.72T + 97T^{2} \) |
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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
−8.614107191948733816645568783015, −7.53185425677244658424527108050, −7.08848199438407137704930335674, −6.32680758854771397587940626865, −5.47130485334760783543479140444, −4.84969681378296958990238150081, −3.98887270171592068336882362086, −3.22381150221374671838013749606, −2.88956830951419598622175527631, −0.54151130280061613618244751313,
0.54151130280061613618244751313, 2.88956830951419598622175527631, 3.22381150221374671838013749606, 3.98887270171592068336882362086, 4.84969681378296958990238150081, 5.47130485334760783543479140444, 6.32680758854771397587940626865, 7.08848199438407137704930335674, 7.53185425677244658424527108050, 8.614107191948733816645568783015
