L(s) = 1 | − i·3-s + (−2 − i)5-s + 2i·7-s − 9-s − 6·11-s + 2i·13-s + (−1 + 2i)15-s − 6i·17-s − 4·19-s + 2·21-s + 8i·23-s + (3 + 4i)25-s + i·27-s − 8·31-s + 6i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s − 1.80·11-s + 0.554i·13-s + (−0.258 + 0.516i)15-s − 1.45i·17-s − 0.917·19-s + 0.436·21-s + 1.66i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s − 1.43·31-s + 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + iT \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−10.66840457670411398822651275117, −9.379779331066272306026624689190, −8.582928521421240605181841527889, −7.70213473430766500337126084682, −7.07048426959383446513857128899, −5.56205252071994952756831807401, −4.92558744211811811881895838567, −3.36303345410697834203407944532, −2.12390147625085523619608381123, 0,
2.62022662343105515247089464590, 3.79001386413395510669916137069, 4.63266527209118222835839540378, 5.85525899680112467077948956502, 7.06682155458296230482510529288, 8.028077444033358273138741072781, 8.531785347225804766760063324655, 10.16540835887823097113391271498, 10.65614914504339678962513405921