| L(s) = 1 | + (−1.51 + 1.06i)2-s + (−1.16 + 0.101i)3-s + (0.485 − 1.33i)4-s + (0.762 + 2.10i)5-s + (1.65 − 1.38i)6-s + (−4.38 − 1.17i)7-s + (−0.278 − 1.03i)8-s + (−1.60 + 0.283i)9-s + (−3.38 − 2.37i)10-s + (−0.761 + 1.31i)11-s + (−0.429 + 1.60i)12-s + (−0.0120 + 0.138i)13-s + (7.89 − 2.87i)14-s + (−1.10 − 2.37i)15-s + (3.69 + 3.10i)16-s + (2.90 + 4.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.749i)2-s + (−0.672 + 0.0588i)3-s + (0.242 − 0.666i)4-s + (0.340 + 0.940i)5-s + (0.675 − 0.567i)6-s + (−1.65 − 0.444i)7-s + (−0.0984 − 0.367i)8-s + (−0.536 + 0.0945i)9-s + (−1.06 − 0.751i)10-s + (−0.229 + 0.397i)11-s + (−0.123 + 0.462i)12-s + (−0.00335 + 0.0383i)13-s + (2.10 − 0.767i)14-s + (−0.284 − 0.612i)15-s + (0.924 + 0.775i)16-s + (0.705 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0175759 - 0.225854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0175759 - 0.225854i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.762 - 2.10i)T \) |
| 19 | \( 1 + (4.14 + 1.34i)T \) |
| good | 2 | \( 1 + (1.51 - 1.06i)T + (0.684 - 1.87i)T^{2} \) |
| 3 | \( 1 + (1.16 - 0.101i)T + (2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (4.38 + 1.17i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.761 - 1.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0120 - 0.138i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.90 - 4.15i)T + (-5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-3.12 - 6.70i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.346 + 1.96i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.08 - 0.625i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.05 - 2.05i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.05 + 1.25i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.801 + 0.373i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (2.02 + 1.41i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (10.1 - 4.74i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (1.86 - 10.5i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.79 + 1.01i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.15 + 11.6i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (1.79 + 4.92i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.526 + 6.01i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-4.43 - 3.72i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.122 - 0.458i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.43 + 1.20i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.975 - 0.682i)T + (33.1 - 91.1i)T^{2} \) |
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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
−14.95663516014069064385476353622, −13.47907830435726223673161798081, −12.48002513512460420489555866140, −10.86453188752939120951787534610, −10.10588190758231932482800749577, −9.245005554753513463931517484803, −7.67099746693219946273499944183, −6.60024854818273050239746270877, −5.96828916738189028841088028052, −3.37288140109145847532716534109,
0.38983785378893631615785717485, 2.83014149894817355760620691656, 5.33056944931417377476744410981, 6.37423441360354239907104713992, 8.401438847403575246499272576718, 9.239261734575159095221282179058, 10.04698089728437242373628227792, 11.17638436864746915866787077140, 12.31703216970731842002451042599, 12.86665306685783968969120033739
