L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (0.866 + 0.5i)23-s − 0.999i·28-s + (1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (0.866 + 0.5i)23-s − 0.999i·28-s + (1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6079343496\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6079343496\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73T + T^{2} \) |
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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
−9.902843052736704120766080394402, −9.155518401249681928513130926407, −8.064151167198344532055962641848, −7.81590633233448395869169405841, −6.63532427232161056470101490968, −5.95474826851794784163257743483, −4.56140998207264830748760057485, −3.47160226851284291616910371761, −2.69880606878090054973636048477, −1.31612938331364370024605107047,
0.74138255448435302885048051543, 2.54372587052313966033813673154, 3.35286231215934185708487548345, 5.02172893201302514963705171709, 5.72505500947506354828497585868, 6.69196782425426827306123981798, 7.07467259178988899058909605317, 8.314252624983341951786152656157, 8.897192729319005586114327657601, 9.609269781558876904826752804907