| L(s) = 1 | + 1.13·2-s + 0.924·3-s − 0.721·4-s − 5-s + 1.04·6-s + 4.16·7-s − 3.07·8-s − 2.14·9-s − 1.13·10-s − 6.34·11-s − 0.667·12-s + 6.73·13-s + 4.70·14-s − 0.924·15-s − 2.03·16-s − 7.24·17-s − 2.42·18-s − 2.67·19-s + 0.721·20-s + 3.85·21-s − 7.17·22-s − 2.48·23-s − 2.84·24-s + 25-s + 7.62·26-s − 4.75·27-s − 3.00·28-s + ⋯ |
| L(s) = 1 | + 0.799·2-s + 0.533·3-s − 0.360·4-s − 0.447·5-s + 0.426·6-s + 1.57·7-s − 1.08·8-s − 0.714·9-s − 0.357·10-s − 1.91·11-s − 0.192·12-s + 1.86·13-s + 1.25·14-s − 0.238·15-s − 0.509·16-s − 1.75·17-s − 0.571·18-s − 0.613·19-s + 0.161·20-s + 0.840·21-s − 1.53·22-s − 0.517·23-s − 0.580·24-s + 0.200·25-s + 1.49·26-s − 0.915·27-s − 0.567·28-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 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 0.924T + 3T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 6.73T + 13T^{2} \) |
| 17 | \( 1 + 7.24T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 - 1.96T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.99T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 0.421T + 97T^{2} \) |
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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
−8.609507008224312517056380168194, −8.182362834681391108540849591664, −7.43788202874538941417609219157, −5.95566609933399187525037293740, −5.52215168339454377448887027583, −4.51309456380655148427940658949, −4.00397112927031210362100646909, −2.89851006128842750158697276120, −1.97664623821202712565611228288, 0,
1.97664623821202712565611228288, 2.89851006128842750158697276120, 4.00397112927031210362100646909, 4.51309456380655148427940658949, 5.52215168339454377448887027583, 5.95566609933399187525037293740, 7.43788202874538941417609219157, 8.182362834681391108540849591664, 8.609507008224312517056380168194
