| L(s) = 1 | − 2-s − 0.0106·3-s + 4-s − 1.19·5-s + 0.0106·6-s + 1.61·7-s − 8-s − 2.99·9-s + 1.19·10-s − 2.53·11-s − 0.0106·12-s + 6.61·13-s − 1.61·14-s + 0.0127·15-s + 16-s − 1.49·17-s + 2.99·18-s + 1.43·19-s − 1.19·20-s − 0.0172·21-s + 2.53·22-s − 0.350·23-s + 0.0106·24-s − 3.57·25-s − 6.61·26-s + 0.0639·27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.00615·3-s + 0.5·4-s − 0.534·5-s + 0.00435·6-s + 0.611·7-s − 0.353·8-s − 0.999·9-s + 0.377·10-s − 0.765·11-s − 0.00307·12-s + 1.83·13-s − 0.432·14-s + 0.00328·15-s + 0.250·16-s − 0.362·17-s + 0.707·18-s + 0.329·19-s − 0.267·20-s − 0.00376·21-s + 0.541·22-s − 0.0731·23-s + 0.00217·24-s − 0.714·25-s − 1.29·26-s + 0.0123·27-s + 0.305·28-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 | \( 1 + 0.0106T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 0.350T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 + 0.851T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 7.90T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−8.074010272904255054158011340707, −7.71943695886720528437234147183, −6.68532107250393106047714061367, −5.86762120696049676069455318273, −5.30548952576324801639477706705, −4.10978839593961841048375365949, −3.34826243811633108786078454189, −2.37018901333246815499644900356, −1.27241568428055522050979245185, 0,
1.27241568428055522050979245185, 2.37018901333246815499644900356, 3.34826243811633108786078454189, 4.10978839593961841048375365949, 5.30548952576324801639477706705, 5.86762120696049676069455318273, 6.68532107250393106047714061367, 7.71943695886720528437234147183, 8.074010272904255054158011340707
