| L(s) = 1 | + 2-s + 2.29·3-s + 4-s − 1.09·5-s + 2.29·6-s − 3.50·7-s + 8-s + 2.26·9-s − 1.09·10-s + 0.822·11-s + 2.29·12-s − 1.82·13-s − 3.50·14-s − 2.50·15-s + 16-s − 6.19·17-s + 2.26·18-s − 4.65·19-s − 1.09·20-s − 8.03·21-s + 0.822·22-s + 5.11·23-s + 2.29·24-s − 3.81·25-s − 1.82·26-s − 1.68·27-s − 3.50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.487·5-s + 0.936·6-s − 1.32·7-s + 0.353·8-s + 0.755·9-s − 0.344·10-s + 0.247·11-s + 0.662·12-s − 0.507·13-s − 0.935·14-s − 0.646·15-s + 0.250·16-s − 1.50·17-s + 0.534·18-s − 1.06·19-s − 0.243·20-s − 1.75·21-s + 0.175·22-s + 1.06·23-s + 0.468·24-s − 0.762·25-s − 0.358·26-s − 0.324·27-s − 0.661·28-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 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 - 0.822T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.10T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 - 0.798T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 - 0.944T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.072570932314491454303743697893, −7.29389912555950458745312875794, −6.64550127702867230811752571274, −6.04266215169828854172652209019, −4.75539566627854175571192871040, −4.08071541764345616824328867300, −3.37004643121664915327783331854, −2.72901541996877468653669671745, −1.95552678524775425740855060587, 0,
1.95552678524775425740855060587, 2.72901541996877468653669671745, 3.37004643121664915327783331854, 4.08071541764345616824328867300, 4.75539566627854175571192871040, 6.04266215169828854172652209019, 6.64550127702867230811752571274, 7.29389912555950458745312875794, 8.072570932314491454303743697893
