L(s) = 1 | − 2-s − 0.267·3-s + 4-s − 0.925·5-s + 0.267·6-s − 3.16·7-s − 8-s − 2.92·9-s + 0.925·10-s + 6.35·11-s − 0.267·12-s + 3.11·13-s + 3.16·14-s + 0.247·15-s + 16-s − 5.68·17-s + 2.92·18-s − 7.58·19-s − 0.925·20-s + 0.845·21-s − 6.35·22-s − 7.47·23-s + 0.267·24-s − 4.14·25-s − 3.11·26-s + 1.58·27-s − 3.16·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.154·3-s + 0.5·4-s − 0.414·5-s + 0.109·6-s − 1.19·7-s − 0.353·8-s − 0.976·9-s + 0.292·10-s + 1.91·11-s − 0.0771·12-s + 0.863·13-s + 0.846·14-s + 0.0638·15-s + 0.250·16-s − 1.37·17-s + 0.690·18-s − 1.73·19-s − 0.207·20-s + 0.184·21-s − 1.35·22-s − 1.55·23-s + 0.0545·24-s − 0.828·25-s − 0.610·26-s + 0.304·27-s − 0.598·28-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
\(L(1)\) |
\(\approx\) |
\(0.6060638473\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6060638473\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.267T + 3T^{2} \) |
| 5 | \( 1 + 0.925T + 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 9.76T + 43T^{2} \) |
| 47 | \( 1 + 9.16T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.13T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−8.626647134669080372619232325550, −7.963501215166330105424359253942, −6.66192363796835584025730492500, −6.30688812695836640577695729724, −6.11723096625311299720471653453, −4.32937060379581851652280569357, −3.91021178666378793934576166588, −2.88340303545953804208951041165, −1.87401960161153660957918439187, −0.48215519749704538373498486231,
0.48215519749704538373498486231, 1.87401960161153660957918439187, 2.88340303545953804208951041165, 3.91021178666378793934576166588, 4.32937060379581851652280569357, 6.11723096625311299720471653453, 6.30688812695836640577695729724, 6.66192363796835584025730492500, 7.963501215166330105424359253942, 8.626647134669080372619232325550