| L(s) = 1 | + (1.39 − 0.221i)2-s + (−0.786 + 1.54i)3-s + (1.90 − 0.618i)4-s + (1.98 − 4.59i)5-s + (−0.756 + 2.32i)6-s + (3.83 − 3.83i)7-s + (2.52 − 1.28i)8-s + (−1.76 − 2.42i)9-s + (1.75 − 6.85i)10-s + (10.4 + 7.62i)11-s + (−0.541 + 3.42i)12-s + (−0.104 − 0.0165i)13-s + (4.51 − 6.20i)14-s + (5.52 + 6.66i)15-s + (3.23 − 2.35i)16-s + (3.65 + 7.17i)17-s + ⋯ |
| L(s) = 1 | + (0.698 − 0.110i)2-s + (−0.262 + 0.514i)3-s + (0.475 − 0.154i)4-s + (0.396 − 0.918i)5-s + (−0.126 + 0.388i)6-s + (0.548 − 0.548i)7-s + (0.315 − 0.160i)8-s + (−0.195 − 0.269i)9-s + (0.175 − 0.685i)10-s + (0.954 + 0.693i)11-s + (−0.0451 + 0.285i)12-s + (−0.00802 − 0.00127i)13-s + (0.322 − 0.443i)14-s + (0.368 + 0.444i)15-s + (0.202 − 0.146i)16-s + (0.215 + 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.18958 - 0.344471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18958 - 0.344471i\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.98 + 4.59i)T \) |
| good | 7 | \( 1 + (-3.83 + 3.83i)T - 49iT^{2} \) |
| 11 | \( 1 + (-10.4 - 7.62i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (0.104 + 0.0165i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-3.65 - 7.17i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (3.30 + 1.07i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (0.273 + 1.72i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (43.2 - 14.0i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (3.47 - 10.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-4.05 + 25.6i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (51.0 - 37.1i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (51.7 + 51.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-59.7 - 30.4i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (28.3 - 55.5i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-39.9 - 55.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-35.3 - 25.7i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (1.08 + 2.13i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (4.80 + 14.7i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (16.2 + 102. i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-31.5 + 10.2i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (89.8 - 45.7i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (60.8 - 83.8i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (60.6 + 30.9i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−12.67015491148734934497598039138, −11.88285144866502523778575726422, −10.81781084646060061706960307267, −9.782536536664411832021570132650, −8.736258466474631581758207212638, −7.24652736225973386304687668219, −5.86730597234414426453418260629, −4.77847688771192112720503228555, −3.89813114617943513291056393857, −1.58447802239305999111459636083,
1.99935968340603676050453398564, 3.52859488928246486813831957572, 5.33211747268474987375860734563, 6.26147932633239005289595381049, 7.20043728810979556585794100239, 8.486543716990600252180530650574, 9.925918243198292387511357531147, 11.43572411948016337582831401711, 11.53547440350121366739586846540, 12.96339683133807980004083291338
