L(s) = 1 | + 2-s − 2.43·3-s + 4-s + 1.19·5-s − 2.43·6-s + 2.99·7-s + 8-s + 2.94·9-s + 1.19·10-s − 6.09·11-s − 2.43·12-s + 2.43·13-s + 2.99·14-s − 2.91·15-s + 16-s − 0.802·17-s + 2.94·18-s + 3.70·19-s + 1.19·20-s − 7.31·21-s − 6.09·22-s − 6.79·23-s − 2.43·24-s − 3.57·25-s + 2.43·26-s + 0.128·27-s + 2.99·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.40·3-s + 0.5·4-s + 0.533·5-s − 0.995·6-s + 1.13·7-s + 0.353·8-s + 0.982·9-s + 0.377·10-s − 1.83·11-s − 0.703·12-s + 0.675·13-s + 0.801·14-s − 0.751·15-s + 0.250·16-s − 0.194·17-s + 0.694·18-s + 0.850·19-s + 0.266·20-s − 1.59·21-s − 1.29·22-s − 1.41·23-s − 0.497·24-s − 0.715·25-s + 0.477·26-s + 0.0247·27-s + 0.566·28-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 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 0.802T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 + 9.18T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 + 1.55T + 41T^{2} \) |
| 43 | \( 1 + 8.87T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 0.385T + 71T^{2} \) |
| 73 | \( 1 + 0.664T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.73479754390023244562045244008, −7.39502182737359450271650551907, −6.14465390832545385718390098745, −5.69146247429316526978980892656, −5.27924117693756785174514097731, −4.61538615207674542500615635374, −3.62657107501270761577459656032, −2.32126081146095369395839110579, −1.53778477962030829400295494957, 0,
1.53778477962030829400295494957, 2.32126081146095369395839110579, 3.62657107501270761577459656032, 4.61538615207674542500615635374, 5.27924117693756785174514097731, 5.69146247429316526978980892656, 6.14465390832545385718390098745, 7.39502182737359450271650551907, 7.73479754390023244562045244008