L(s) = 1 | − 2-s − 0.941·3-s + 4-s − 0.129·5-s + 0.941·6-s + 0.350·7-s − 8-s − 2.11·9-s + 0.129·10-s + 3.66·11-s − 0.941·12-s − 0.342·13-s − 0.350·14-s + 0.122·15-s + 16-s − 0.324·17-s + 2.11·18-s − 0.916·19-s − 0.129·20-s − 0.330·21-s − 3.66·22-s + 1.25·23-s + 0.941·24-s − 4.98·25-s + 0.342·26-s + 4.81·27-s + 0.350·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.543·3-s + 0.5·4-s − 0.0580·5-s + 0.384·6-s + 0.132·7-s − 0.353·8-s − 0.704·9-s + 0.0410·10-s + 1.10·11-s − 0.271·12-s − 0.0950·13-s − 0.0936·14-s + 0.0315·15-s + 0.250·16-s − 0.0787·17-s + 0.498·18-s − 0.210·19-s − 0.0290·20-s − 0.0720·21-s − 0.782·22-s + 0.260·23-s + 0.192·24-s − 0.996·25-s + 0.0671·26-s + 0.926·27-s + 0.0662·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 0.941T + 3T^{2} \) |
| 5 | \( 1 + 0.129T + 5T^{2} \) |
| 7 | \( 1 - 0.350T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 0.342T + 13T^{2} \) |
| 17 | \( 1 + 0.324T + 17T^{2} \) |
| 19 | \( 1 + 0.916T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 8.36T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 + 8.00T + 43T^{2} \) |
| 47 | \( 1 + 2.84T + 47T^{2} \) |
| 53 | \( 1 - 7.31T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 5.69T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.47T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.293509190203989313225220955184, −7.31624233206908236194968170568, −6.64549006855110791232670068864, −5.98861634172547277951322761119, −5.28010985734822988991017366369, −4.24866340631370994936786425267, −3.35581691948260380144503998827, −2.27442615285556702615792180700, −1.22214239555870270671850095324, 0,
1.22214239555870270671850095324, 2.27442615285556702615792180700, 3.35581691948260380144503998827, 4.24866340631370994936786425267, 5.28010985734822988991017366369, 5.98861634172547277951322761119, 6.64549006855110791232670068864, 7.31624233206908236194968170568, 8.293509190203989313225220955184