| L(s) = 1 | + 2.34·2-s − 2.94·3-s + 3.47·4-s + 5-s − 6.88·6-s − 2.44·7-s + 3.45·8-s + 5.65·9-s + 2.34·10-s − 0.203·11-s − 10.2·12-s + 0.00124·13-s − 5.72·14-s − 2.94·15-s + 1.14·16-s + 0.0897·17-s + 13.2·18-s − 5.88·19-s + 3.47·20-s + 7.20·21-s − 0.475·22-s − 1.69·23-s − 10.1·24-s + 25-s + 0.00292·26-s − 7.81·27-s − 8.51·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 1.69·3-s + 1.73·4-s + 0.447·5-s − 2.81·6-s − 0.925·7-s + 1.22·8-s + 1.88·9-s + 0.740·10-s − 0.0613·11-s − 2.95·12-s + 0.000346·13-s − 1.53·14-s − 0.759·15-s + 0.285·16-s + 0.0217·17-s + 3.12·18-s − 1.35·19-s + 0.777·20-s + 1.57·21-s − 0.101·22-s − 0.353·23-s − 2.07·24-s + 0.200·25-s + 0.000573·26-s − 1.50·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 0.203T + 11T^{2} \) |
| 13 | \( 1 - 0.00124T + 13T^{2} \) |
| 17 | \( 1 - 0.0897T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 - 6.98T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 + 9.06T + 67T^{2} \) |
| 71 | \( 1 + 9.34T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−8.888621023038077310088502621677, −7.36112347199393400178382870189, −6.58692693546664888057381277751, −6.15400508989295000637990632511, −5.63271837437311359747637762502, −4.78868145527075994660184150536, −4.14194835018823440827043255307, −3.08689536196847354134531450913, −1.82595690906072820104338028785, 0,
1.82595690906072820104338028785, 3.08689536196847354134531450913, 4.14194835018823440827043255307, 4.78868145527075994660184150536, 5.63271837437311359747637762502, 6.15400508989295000637990632511, 6.58692693546664888057381277751, 7.36112347199393400178382870189, 8.888621023038077310088502621677
