| L(s) = 1 | + (1.39 + 0.201i)2-s + (−1.02 + 1.39i)3-s + (1.91 + 0.562i)4-s + (−2.12 + 2.53i)5-s + (−1.72 + 1.74i)6-s + (1.10 − 3.02i)7-s + (2.57 + 1.17i)8-s + (−0.879 − 2.86i)9-s + (−3.48 + 3.11i)10-s + (2.39 − 2.01i)11-s + (−2.75 + 2.09i)12-s + (0.431 − 2.44i)13-s + (2.15 − 4.01i)14-s + (−1.33 − 5.56i)15-s + (3.36 + 2.16i)16-s + (1.16 + 0.672i)17-s + ⋯ |
| L(s) = 1 | + (0.989 + 0.142i)2-s + (−0.594 + 0.804i)3-s + (0.959 + 0.281i)4-s + (−0.950 + 1.13i)5-s + (−0.702 + 0.711i)6-s + (0.416 − 1.14i)7-s + (0.909 + 0.414i)8-s + (−0.293 − 0.956i)9-s + (−1.10 + 0.986i)10-s + (0.723 − 0.606i)11-s + (−0.796 + 0.604i)12-s + (0.119 − 0.678i)13-s + (0.575 − 1.07i)14-s + (−0.345 − 1.43i)15-s + (0.841 + 0.540i)16-s + (0.282 + 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22420 + 0.672455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22420 + 0.672455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.201i)T \) |
| 3 | \( 1 + (1.02 - 1.39i)T \) |
| good | 5 | \( 1 + (2.12 - 2.53i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 3.02i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.39 + 2.01i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.431 + 2.44i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 0.672i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.00 - 2.88i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.02 - 0.736i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (7.98 - 1.40i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.43 + 3.93i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.857 + 1.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.757 - 0.133i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.738 - 0.880i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.75 + 1.00i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 2.35iT - 53T^{2} \) |
| 59 | \( 1 + (3.53 + 2.96i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 0.390i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-14.0 - 2.47i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.55 - 7.88i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.23 - 10.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.28 - 1.63i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.863 + 4.89i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-5.96 + 3.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.19 - 4.35i)T + (16.8 - 95.5i)T^{2} \) |
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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
−14.30810647482368222087119220475, −12.80392955425448672077994295077, −11.46302008308099624577300354603, −11.05796198495284197743168012402, −10.23185478817461298555891720289, −8.032233463954405018933666422513, −6.94557274143657066159947662555, −5.82557671615039765136073301674, −4.09741006066496806786801901273, −3.57619581354226467873524282517,
1.89197246400016204712587603627, 4.30899839212207387437290006222, 5.29100512029357918300437920508, 6.56424525942106810551231647891, 7.83893252337647154685290743322, 9.029822537743841228512778894350, 11.09181201828188605908197519803, 11.91840735657662130753380577495, 12.32323128250191575731362546367, 13.17338920949291791754895103245
