| L(s) = 1 | + (−1.39 + 0.221i)2-s + (0.786 − 1.54i)3-s + (1.90 − 0.618i)4-s + (3.79 − 3.25i)5-s + (−0.756 + 2.32i)6-s + (0.904 − 0.904i)7-s + (−2.52 + 1.28i)8-s + (−1.76 − 2.42i)9-s + (−4.58 + 5.38i)10-s + (−3.26 − 2.37i)11-s + (0.541 − 3.42i)12-s + (4.03 + 0.639i)13-s + (−1.06 + 1.46i)14-s + (−2.02 − 8.41i)15-s + (3.23 − 2.35i)16-s + (−12.9 − 25.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (0.262 − 0.514i)3-s + (0.475 − 0.154i)4-s + (0.759 − 0.650i)5-s + (−0.126 + 0.388i)6-s + (0.129 − 0.129i)7-s + (−0.315 + 0.160i)8-s + (−0.195 − 0.269i)9-s + (−0.458 + 0.538i)10-s + (−0.296 − 0.215i)11-s + (0.0451 − 0.285i)12-s + (0.310 + 0.0491i)13-s + (−0.0759 + 0.104i)14-s + (−0.135 − 0.561i)15-s + (0.202 − 0.146i)16-s + (−0.760 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02272 - 0.717641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02272 - 0.717641i\) |
| \(L(2)\) |
|
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 | \( 1 + (-3.79 + 3.25i)T \) |
| good | 7 | \( 1 + (-0.904 + 0.904i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.26 + 2.37i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-4.03 - 0.639i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (12.9 + 25.3i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-26.0 - 8.45i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (3.54 + 22.4i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (-27.9 + 9.09i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (12.2 - 37.8i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 9.69i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (37.0 - 26.9i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-9.68 - 9.68i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-26.7 - 13.6i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (28.0 - 55.0i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-23.1 - 31.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (12.1 + 8.83i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-24.5 - 48.1i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-43.0 - 132. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (14.4 + 91.1i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-63.1 + 20.5i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (48.0 - 24.4i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-69.8 + 96.0i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-93.1 - 47.4i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−12.56493981204812208395165744504, −11.59457377768816916837105676677, −10.34474326315928802873980923504, −9.316252531823187088792069993008, −8.536595900057760947039247832813, −7.40418138502828841078171949394, −6.25806663808854691982934005420, −4.96546696368344773444776603734, −2.68721212472445186289047471052, −1.05916353366661809111895154711,
2.02129749388748381423847016328, 3.49817031909119254197871170816, 5.39536097476769074085372989854, 6.63740995715919308973362763907, 7.902976841640781338454679839189, 9.043301431194378463582152703665, 9.926147210960431013478157376384, 10.70519300110792032422412070478, 11.61607174278637823296546915524, 13.09530755105249542727516777104
