L(s) = 1 | − 2-s − 1.16·3-s + 4-s + 3.05·5-s + 1.16·6-s + 2.13·7-s − 8-s − 1.64·9-s − 3.05·10-s + 2.96·11-s − 1.16·12-s − 2.55·13-s − 2.13·14-s − 3.55·15-s + 16-s + 0.966·17-s + 1.64·18-s − 1.03·19-s + 3.05·20-s − 2.48·21-s − 2.96·22-s − 7.18·23-s + 1.16·24-s + 4.35·25-s + 2.55·26-s + 5.40·27-s + 2.13·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.671·3-s + 0.5·4-s + 1.36·5-s + 0.474·6-s + 0.807·7-s − 0.353·8-s − 0.549·9-s − 0.966·10-s + 0.893·11-s − 0.335·12-s − 0.708·13-s − 0.571·14-s − 0.918·15-s + 0.250·16-s + 0.234·17-s + 0.388·18-s − 0.236·19-s + 0.683·20-s − 0.542·21-s − 0.632·22-s − 1.49·23-s + 0.237·24-s + 0.870·25-s + 0.501·26-s + 1.04·27-s + 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 - 0.966T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 5.13T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.34T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.88T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 + 0.694T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−8.224823017782181898111830804759, −7.32831152772085139517840989867, −6.49100198083967446439469581385, −5.91387335733246083296374485859, −5.35911390527497015112284712825, −4.49294814359875909736209279943, −3.19274079067380770995775457032, −1.99287556329014184697603943368, −1.56054190415359343010079737959, 0,
1.56054190415359343010079737959, 1.99287556329014184697603943368, 3.19274079067380770995775457032, 4.49294814359875909736209279943, 5.35911390527497015112284712825, 5.91387335733246083296374485859, 6.49100198083967446439469581385, 7.32831152772085139517840989867, 8.224823017782181898111830804759