| L(s) = 1 | + (−3.28 − 0.879i)2-s + (2.65 − 1.40i)3-s + (6.54 + 3.78i)4-s + (1.19 − 9.04i)5-s + (−9.94 + 2.26i)6-s + (−0.387 + 0.0509i)7-s + (−8.55 − 8.55i)8-s + (5.07 − 7.43i)9-s + (−11.8 + 28.6i)10-s + (6.56 − 0.863i)11-s + (22.6 + 0.855i)12-s + (6.54 + 3.77i)13-s + (1.31 + 0.173i)14-s + (−9.51 − 25.6i)15-s + (5.45 + 9.45i)16-s + (0.857 + 16.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.64 − 0.439i)2-s + (0.884 − 0.466i)3-s + (1.63 + 0.945i)4-s + (0.238 − 1.80i)5-s + (−1.65 + 0.377i)6-s + (−0.0553 + 0.00728i)7-s + (−1.06 − 1.06i)8-s + (0.563 − 0.825i)9-s + (−1.18 + 2.86i)10-s + (0.596 − 0.0785i)11-s + (1.88 + 0.0713i)12-s + (0.503 + 0.290i)13-s + (0.0940 + 0.0123i)14-s + (−0.634 − 1.71i)15-s + (0.341 + 0.590i)16-s + (0.0504 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.397776 - 0.835837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.397776 - 0.835837i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.65 + 1.40i)T \) |
| 17 | \( 1 + (-0.857 - 16.9i)T \) |
| good | 2 | \( 1 + (3.28 + 0.879i)T + (3.46 + 2i)T^{2} \) |
| 5 | \( 1 + (-1.19 + 9.04i)T + (-24.1 - 6.47i)T^{2} \) |
| 7 | \( 1 + (0.387 - 0.0509i)T + (47.3 - 12.6i)T^{2} \) |
| 11 | \( 1 + (-6.56 + 0.863i)T + (116. - 31.3i)T^{2} \) |
| 13 | \( 1 + (-6.54 - 3.77i)T + (84.5 + 146. i)T^{2} \) |
| 19 | \( 1 + (3.31 + 3.31i)T + 361iT^{2} \) |
| 23 | \( 1 + (11.7 + 15.3i)T + (-136. + 510. i)T^{2} \) |
| 29 | \( 1 + (36.7 + 28.2i)T + (217. + 812. i)T^{2} \) |
| 31 | \( 1 + (3.89 - 29.5i)T + (-928. - 248. i)T^{2} \) |
| 37 | \( 1 + (-54.5 + 22.5i)T + (968. - 968. i)T^{2} \) |
| 41 | \( 1 + (22.9 - 17.5i)T + (435. - 1.62e3i)T^{2} \) |
| 43 | \( 1 + (-74.0 - 19.8i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-3.24 - 5.62i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-0.227 + 0.227i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (23.3 - 6.24i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (2.27 - 0.299i)T + (3.59e3 - 963. i)T^{2} \) |
| 67 | \( 1 + (-41.5 + 71.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-47.7 - 115. i)T + (-3.56e3 + 3.56e3i)T^{2} \) |
| 73 | \( 1 + (-43.7 - 105. i)T + (-3.76e3 + 3.76e3i)T^{2} \) |
| 79 | \( 1 + (1.90 + 14.4i)T + (-6.02e3 + 1.61e3i)T^{2} \) |
| 83 | \( 1 + (-32.3 - 8.65i)T + (5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 - 75.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + (73.7 - 96.1i)T + (-2.43e3 - 9.08e3i)T^{2} \) |
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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
−12.41757836410994743415295862365, −11.27320991594715904466630059259, −9.777391585036773855175616545821, −9.159776937118897097442057458263, −8.476206464994565613472629194715, −7.80293321006781904362782149920, −6.23371338383692246084474639255, −4.09994206440958218351452801790, −1.99549185963076317483676453993, −0.962424576395809601714442124162,
2.13768778365975417272189622120, 3.51787509709356123192584735505, 6.11310386456634143132685803562, 7.21597625428936530269351382142, 7.82221164668926150886778430625, 9.214239982708710036894688908717, 9.775180094309934901711791051406, 10.70990828078639414150928736019, 11.35472579770039284317885069056, 13.50675249810626175524674581952
