| L(s) = 1 | + (1.65 + 1.12i)2-s + (0.733 + 0.680i)3-s + (0.729 + 1.85i)4-s + (−0.400 + 0.123i)5-s + (0.444 + 1.94i)6-s + (−1.90 − 1.83i)7-s + (0.00157 − 0.00688i)8-s + (0.0747 + 0.997i)9-s + (−0.800 − 0.246i)10-s + (0.0436 − 0.582i)11-s + (−0.729 + 1.85i)12-s + (−3.22 + 1.55i)13-s + (−1.08 − 5.17i)14-s + (−0.377 − 0.181i)15-s + (2.93 − 2.72i)16-s + (−0.518 + 0.0780i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.796i)2-s + (0.423 + 0.392i)3-s + (0.364 + 0.929i)4-s + (−0.178 + 0.0552i)5-s + (0.181 + 0.795i)6-s + (−0.721 − 0.692i)7-s + (0.000555 − 0.00243i)8-s + (0.0249 + 0.332i)9-s + (−0.252 − 0.0780i)10-s + (0.0131 − 0.175i)11-s + (−0.210 + 0.536i)12-s + (−0.895 + 0.431i)13-s + (−0.291 − 1.38i)14-s + (−0.0974 − 0.0469i)15-s + (0.734 − 0.681i)16-s + (−0.125 + 0.0189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66694 + 1.02086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66694 + 1.02086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.733 - 0.680i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
| good | 2 | \( 1 + (-1.65 - 1.12i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (0.400 - 0.123i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.0436 + 0.582i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (3.22 - 1.55i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.518 - 0.0780i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.603 + 1.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.89 - 1.34i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.11 + 2.65i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (3.57 - 6.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.29 + 3.29i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.789 + 3.46i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.23 - 5.42i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (4.74 + 3.23i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.18 + 3.01i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-8.24 - 2.54i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.07 + 5.29i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (3.72 - 6.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.36 - 6.73i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (10.4 - 7.12i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (4.12 + 7.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-15.5 - 7.50i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.281 + 3.75i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 5.47T + 97T^{2} \) |
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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
−13.33081568891512291798018921890, −12.79149150674891637260718382603, −11.38135694030524287335100916174, −10.09654233411440178666544553226, −9.122359178022479970768970287614, −7.47033317675065128481809108809, −6.82100270101470405523143582610, −5.38186839879301661559586575524, −4.25738183292805251825711441155, −3.19600883023837074263905664223,
2.34220150382270077224051823681, 3.37703470503471094682354100964, 4.83687054989222948893830647336, 6.06142554821900380395010728367, 7.49000632937399259960170889724, 8.829585649800559619777690508571, 9.981691624934872511817829562097, 11.27031468224780046583311056317, 12.23010462156433499855752372276, 12.83098044355940316039402247366
