L(s) = 1 | + i·2-s + (−0.151 + 1.72i)3-s − 4-s + (0.866 − 0.5i)5-s + (−1.72 − 0.151i)6-s + (−1.64 + 2.85i)7-s − i·8-s + (−2.95 − 0.522i)9-s + (0.5 + 0.866i)10-s + (−2.34 − 4.05i)11-s + (0.151 − 1.72i)12-s + (−3.57 + 2.06i)13-s + (−2.85 − 1.64i)14-s + (0.731 + 1.56i)15-s + 16-s + (3.02 − 5.23i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.0874 + 0.996i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.704 − 0.0618i)6-s + (−0.622 + 1.07i)7-s − 0.353i·8-s + (−0.984 − 0.174i)9-s + (0.158 + 0.273i)10-s + (−0.706 − 1.22i)11-s + (0.0437 − 0.498i)12-s + (−0.991 + 0.572i)13-s + (−0.762 − 0.440i)14-s + (0.188 + 0.405i)15-s + 0.250·16-s + (0.732 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0488 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0488 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0783206 - 0.0745868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0783206 - 0.0745868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.151 - 1.72i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (5.16 - 2.07i)T \) |
good | 7 | \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.34 + 4.05i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.57 - 2.06i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 5.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.11 - 1.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.244T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 37 | \( 1 + (7.39 + 4.27i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.83 - 3.94i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 5.79i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.99iT - 47T^{2} \) |
| 53 | \( 1 + (0.996 + 1.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.641 - 0.370i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 6.30iT - 61T^{2} \) |
| 67 | \( 1 + (-1.90 - 3.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.36 - 4.83i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.290 + 0.167i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.1 + 6.42i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.42 - 2.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 0.149T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−10.41794744775266113777720396997, −9.625773597827496647813302749990, −9.101634830181089951026781496327, −8.428795442480363968203768413384, −7.31219765466141609296419870478, −6.09868543436206373623499831736, −5.48743742343809360045601837466, −4.92603151554678386639644912779, −3.51002570625187074674709289629, −2.59699409226469021143303327263,
0.04849908149598621570513162272, 1.64446063353012368700184683478, 2.62832111459531948707125992731, 3.74252864552856085984232818822, 5.02679850257588128278755199605, 5.96260109757720396557542228840, 7.22372435449030552352750320262, 7.42163824175202717353166941290, 8.595199168788589707707326111061, 9.770497372321864155807700781104