L(s) = 1 | + 3.37·5-s − 1.37·7-s − 11-s − 5.37·13-s + 0.372·17-s − 6.37·19-s + 5.37·23-s + 6.37·25-s − 1.37·29-s + 0.627·31-s − 4.62·35-s + 2.74·37-s − 0.255·41-s − 9.74·43-s − 1.37·47-s − 5.11·49-s + 10.7·53-s − 3.37·55-s − 7·59-s + 3.37·61-s − 18.1·65-s − 7.74·67-s − 4·71-s + 5.11·73-s + 1.37·77-s − 0.627·79-s − 15.3·83-s + ⋯ |
L(s) = 1 | + 1.50·5-s − 0.518·7-s − 0.301·11-s − 1.49·13-s + 0.0902·17-s − 1.46·19-s + 1.12·23-s + 1.27·25-s − 0.254·29-s + 0.112·31-s − 0.782·35-s + 0.451·37-s − 0.0398·41-s − 1.48·43-s − 0.200·47-s − 0.730·49-s + 1.47·53-s − 0.454·55-s − 0.911·59-s + 0.431·61-s − 2.24·65-s − 0.946·67-s − 0.474·71-s + 0.598·73-s + 0.156·77-s − 0.0706·79-s − 1.68·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 0.627T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 + 0.255T + 41T^{2} \) |
| 43 | \( 1 + 9.74T + 43T^{2} \) |
| 47 | \( 1 + 1.37T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + 0.627T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 9.74T + 97T^{2} \) |
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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
−7.84009843017776050510162243778, −6.86268118760155579687912630363, −6.54061077805822248954601543398, −5.61039047914606034098155311323, −5.09720248416829243112674753578, −4.26733118436619383989946041498, −2.96477905847859692682043104667, −2.42944332300122310886360353692, −1.55108157117675645149728519860, 0,
1.55108157117675645149728519860, 2.42944332300122310886360353692, 2.96477905847859692682043104667, 4.26733118436619383989946041498, 5.09720248416829243112674753578, 5.61039047914606034098155311323, 6.54061077805822248954601543398, 6.86268118760155579687912630363, 7.84009843017776050510162243778