L(s) = 1 | + (0.662 − 0.382i)2-s + (−0.207 + 0.358i)4-s + (−0.5 − 0.866i)5-s + 1.08i·8-s + (−0.662 − 0.382i)10-s + (0.207 + 0.358i)16-s + (0.707 − 1.22i)17-s + (0.662 − 0.382i)19-s + 0.414·20-s + (1.60 − 0.923i)23-s + (−0.499 + 0.866i)25-s + (1.60 + 0.923i)31-s + (−0.662 − 0.382i)32-s − 1.08i·34-s + (0.292 − 0.507i)38-s + ⋯ |
L(s) = 1 | + (0.662 − 0.382i)2-s + (−0.207 + 0.358i)4-s + (−0.5 − 0.866i)5-s + 1.08i·8-s + (−0.662 − 0.382i)10-s + (0.207 + 0.358i)16-s + (0.707 − 1.22i)17-s + (0.662 − 0.382i)19-s + 0.414·20-s + (1.60 − 0.923i)23-s + (−0.499 + 0.866i)25-s + (1.60 + 0.923i)31-s + (−0.662 − 0.382i)32-s − 1.08i·34-s + (0.292 − 0.507i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.469326402\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469326402\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.896861412978410946098197854957, −8.587652087906126322807901544705, −7.60335622230676097720822645499, −6.99248105187843194169020216858, −5.62011425120052498774489791330, −4.87947148932324205178454279735, −4.49097931481786768876079812151, −3.31177383006361436259001046724, −2.71977794140026814598945374003, −1.04275074505802050413646822424,
1.28815609205996080573355440423, 2.93652270411336166320326018032, 3.67772346108254064979208954257, 4.48732434078905049913483692455, 5.46718888755782759466496670917, 6.13127878150313314440968745449, 6.86943736907528620631847851503, 7.61628330307571490131583922745, 8.366718253171957871965968892528, 9.515832681623331537092954678039