| L(s) = 1 | − 0.431·2-s + 0.385·3-s − 1.81·4-s − 0.726·5-s − 0.166·6-s + 1.71·7-s + 1.64·8-s − 2.85·9-s + 0.313·10-s − 1.66·11-s − 0.698·12-s + 2.26·13-s − 0.740·14-s − 0.279·15-s + 2.91·16-s + 17-s + 1.23·18-s + 6.41·19-s + 1.31·20-s + 0.660·21-s + 0.718·22-s − 7.39·23-s + 0.634·24-s − 4.47·25-s − 0.980·26-s − 2.25·27-s − 3.10·28-s + ⋯ |
| L(s) = 1 | − 0.305·2-s + 0.222·3-s − 0.906·4-s − 0.324·5-s − 0.0679·6-s + 0.648·7-s + 0.582·8-s − 0.950·9-s + 0.0991·10-s − 0.501·11-s − 0.201·12-s + 0.629·13-s − 0.197·14-s − 0.0722·15-s + 0.728·16-s + 0.242·17-s + 0.290·18-s + 1.47·19-s + 0.294·20-s + 0.144·21-s + 0.153·22-s − 1.54·23-s + 0.129·24-s − 0.894·25-s − 0.192·26-s − 0.433·27-s − 0.587·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 0.431T + 2T^{2} \) |
| 3 | \( 1 - 0.385T + 3T^{2} \) |
| 5 | \( 1 + 0.726T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 + 9.94T + 31T^{2} \) |
| 37 | \( 1 + 0.591T + 37T^{2} \) |
| 41 | \( 1 + 0.659T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 + 0.401T + 53T^{2} \) |
| 61 | \( 1 - 0.400T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.446829996733992939168327254973, −8.690210252538551878186405488898, −7.916402128759974648725729017345, −7.54372479226093138646658292136, −5.79211730232744558402371278942, −5.33572039101476255109414766615, −4.10220524670824128964148277825, −3.30353654542815962051557975014, −1.70400679602482255777587489213, 0,
1.70400679602482255777587489213, 3.30353654542815962051557975014, 4.10220524670824128964148277825, 5.33572039101476255109414766615, 5.79211730232744558402371278942, 7.54372479226093138646658292136, 7.916402128759974648725729017345, 8.690210252538551878186405488898, 9.446829996733992939168327254973
