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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2790852567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2790852567\) |
\(L(1)\) |
|
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
−8.919139133704940359508044584739, −8.116978465798933485944242320343, −7.72494620575313057018065503122, −6.97392183711798731810302523775, −5.86017332948490486815194469065, −5.01777400816413318875672184481, −4.01645583778648129291401857870, −3.43196080699392321817788825507, −1.79118898187461401334709114161, −0.24257301488232730003929479775,
1.59268658276448261233638046374, 2.57878942768636255745391761225, 3.70942764215946022832579456529, 4.53674002666412140758275106086, 5.73689676463289511180115797185, 6.32007469111780652545898933467, 7.32779163773920255589261285846, 8.191534733769667782684800744270, 8.598324742721025425223543834468, 9.680113595858593646877848130497