| L(s) = 1 | + (−0.647 + 0.210i)2-s + (0.554 − 2.94i)3-s + (−2.86 + 2.07i)4-s + (6.17 + 2.00i)5-s + (0.261 + 2.02i)6-s + (−7.57 + 5.50i)7-s + (3.01 − 4.14i)8-s + (−8.38 − 3.26i)9-s − 4.41·10-s + (4.54 + 9.58i)12-s + (−1.28 − 3.96i)13-s + (3.74 − 5.15i)14-s + (9.33 − 17.0i)15-s + (3.29 − 10.1i)16-s + (−28.4 − 9.24i)17-s + (6.11 + 0.351i)18-s + ⋯ |
| L(s) = 1 | + (−0.323 + 0.105i)2-s + (0.184 − 0.982i)3-s + (−0.715 + 0.519i)4-s + (1.23 + 0.401i)5-s + (0.0435 + 0.337i)6-s + (−1.08 + 0.786i)7-s + (0.376 − 0.518i)8-s + (−0.931 − 0.363i)9-s − 0.441·10-s + (0.378 + 0.799i)12-s + (−0.0991 − 0.305i)13-s + (0.267 − 0.368i)14-s + (0.622 − 1.13i)15-s + (0.205 − 0.633i)16-s + (−1.67 − 0.543i)17-s + (0.339 + 0.0195i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0184083 - 0.208728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0184083 - 0.208728i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.554 + 2.94i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.647 - 0.210i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (-6.17 - 2.00i)T + (20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (7.57 - 5.50i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (1.28 + 3.96i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (28.4 + 9.24i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (8.32 + 6.04i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + 27.8iT - 529T^{2} \) |
| 29 | \( 1 + (-9.49 - 13.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (5.45 + 16.7i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (11.7 - 8.55i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (20.5 - 28.3i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + 64.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (6.14 - 8.45i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (57.4 - 18.6i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (23.9 + 32.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-7.26 + 22.3i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 30.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + (0.946 + 0.307i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-75.6 + 54.9i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (34.9 + 107. i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-98.9 - 32.1i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 20.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-58.7 - 180. i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−10.61037738926721244504238136554, −9.483396712073298390258507125966, −9.026806130814123819911435896162, −8.114281921014148964641178036233, −6.62133045374452466901782871497, −6.44381615687477596719639237544, −4.97948507421018109987253203057, −3.10303020750717485327274482641, −2.21736865785253278560373571226, −0.096689101308202635824924369001,
1.90350888987748586669954868451, 3.68474658711472539157677012674, 4.70503481908191265761891928823, 5.70409442085099536395935169053, 6.65287405625277533588076609854, 8.446041496593340341279519732490, 9.178456933368810887239401148261, 9.825509639129955815970518805803, 10.29838646453752010916909335090, 11.18498910067053134948263303614
