L(s) = 1 | + 2-s + 4-s + (1.28 − 1.83i)5-s + 2.06i·7-s + 8-s + (1.28 − 1.83i)10-s − 3.77·11-s − 2.88·13-s + 2.06i·14-s + 16-s − 5.80·17-s + 0.157i·19-s + (1.28 − 1.83i)20-s − 3.77·22-s − 5.41·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.573 − 0.819i)5-s + 0.779i·7-s + 0.353·8-s + (0.405 − 0.579i)10-s − 1.13·11-s − 0.799·13-s + 0.551i·14-s + 0.250·16-s − 1.40·17-s + 0.0360i·19-s + (0.286 − 0.409i)20-s − 0.805·22-s − 1.12·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6960107464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6960107464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.28 + 1.83i)T \) |
| 37 | \( 1 + (4.80 + 3.72i)T \) |
good | 7 | \( 1 - 2.06iT - 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 0.157iT - 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 0.425iT - 31T^{2} \) |
| 41 | \( 1 + 0.923T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.676iT - 47T^{2} \) |
| 53 | \( 1 + 9.87iT - 53T^{2} \) |
| 59 | \( 1 + 8.47iT - 59T^{2} \) |
| 61 | \( 1 + 1.23iT - 61T^{2} \) |
| 67 | \( 1 - 6.45iT - 67T^{2} \) |
| 71 | \( 1 - 3.28T + 71T^{2} \) |
| 73 | \( 1 + 0.980iT - 73T^{2} \) |
| 79 | \( 1 + 8.04iT - 79T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.344778831726289435342373264616, −7.55933293633569558557878719685, −6.54326703423530421655946484962, −5.88732648058453645848006156866, −5.08477898764206551315859191571, −4.74927577247976804915478193143, −3.62408431398485546455747510490, −2.21223812875443969815376098001, −2.17839496552638751684542141293, −0.13681034262919523107425072641,
1.81446846346585137559333345416, 2.59261509869265145186528654973, 3.40107570339973344148057322204, 4.42010851039706481848394733434, 5.09911083589396483051445831902, 5.94892571176626848005344993959, 6.76406186829740657637299217283, 7.23455344182031902265198192111, 7.980222679708949070943087878462, 8.974679298009982224802428901167