| L(s) = 1 | + (1.39 + 0.221i)2-s + (−0.786 − 1.54i)3-s + (1.90 + 0.618i)4-s + (1.86 − 4.64i)5-s + (−0.756 − 2.32i)6-s + (−6.90 − 6.90i)7-s + (2.52 + 1.28i)8-s + (−1.76 + 2.42i)9-s + (3.62 − 6.07i)10-s + (7.27 − 5.28i)11-s + (−0.541 − 3.42i)12-s + (2.54 − 0.402i)13-s + (−8.11 − 11.1i)14-s + (−8.62 + 0.776i)15-s + (3.23 + 2.35i)16-s + (−10.3 + 20.2i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (−0.262 − 0.514i)3-s + (0.475 + 0.154i)4-s + (0.372 − 0.928i)5-s + (−0.126 − 0.388i)6-s + (−0.985 − 0.985i)7-s + (0.315 + 0.160i)8-s + (−0.195 + 0.269i)9-s + (0.362 − 0.607i)10-s + (0.660 − 0.480i)11-s + (−0.0451 − 0.285i)12-s + (0.195 − 0.0309i)13-s + (−0.579 − 0.797i)14-s + (−0.575 + 0.0517i)15-s + (0.202 + 0.146i)16-s + (−0.607 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.55341 - 1.13920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55341 - 1.13920i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.221i)T \) |
| 3 | \( 1 + (0.786 + 1.54i)T \) |
| 5 | \( 1 + (-1.86 + 4.64i)T \) |
| good | 7 | \( 1 + (6.90 + 6.90i)T + 49iT^{2} \) |
| 11 | \( 1 + (-7.27 + 5.28i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 0.402i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (10.3 - 20.2i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-33.9 + 11.0i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-1.78 + 11.2i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (6.87 + 2.23i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-16.0 - 49.3i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-10.7 - 68.0i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (2.12 + 1.54i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-37.8 + 37.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (75.0 - 38.2i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 20.6i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (10.2 - 14.1i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-4.45 + 3.24i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-50.1 + 98.3i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (0.158 - 0.489i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-7.81 + 49.3i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-95.1 - 30.9i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (48.4 + 24.6i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-18.1 - 25.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (32.7 - 16.6i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.76728600630182284107761066065, −11.90105141027003598789130038762, −10.71205739964962876160928117518, −9.525992726499789923907796949263, −8.273989096003675374856960207672, −6.87734750777576869393578268014, −6.11771460331514973084060137192, −4.76929078514690426953566150122, −3.38841233089326343798391066579, −1.14918461324200968690235933707,
2.56951956420132603973224626747, 3.70450105777637194835330409970, 5.39874483580349122358436825011, 6.25110390813492019876588463493, 7.31030617340581206740814939503, 9.466004976822271303172241588500, 9.724495890331199357332257479502, 11.28980821399717211777996301292, 11.80925774106958667927341612908, 13.01473834545162943573261319000
