| L(s) = 1 | + (−1.39 + 0.204i)2-s + (−1.70 + 0.304i)3-s + (1.91 − 0.572i)4-s + (−2.29 + 1.04i)5-s + (2.32 − 0.774i)6-s + (0.382 + 0.401i)7-s + (−2.56 + 1.19i)8-s + (2.81 − 1.03i)9-s + (2.99 − 1.93i)10-s + (−3.69 − 0.352i)11-s + (−3.09 + 1.55i)12-s + (−1.78 − 0.343i)13-s + (−0.617 − 0.483i)14-s + (3.58 − 2.48i)15-s + (3.34 − 2.19i)16-s + (1.28 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 + 0.144i)2-s + (−0.984 + 0.175i)3-s + (0.958 − 0.286i)4-s + (−1.02 + 0.467i)5-s + (0.948 − 0.316i)6-s + (0.144 + 0.151i)7-s + (−0.906 + 0.422i)8-s + (0.938 − 0.345i)9-s + (0.946 − 0.611i)10-s + (−1.11 − 0.106i)11-s + (−0.892 + 0.450i)12-s + (−0.494 − 0.0953i)13-s + (−0.165 − 0.129i)14-s + (0.926 − 0.640i)15-s + (0.835 − 0.548i)16-s + (0.312 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.364513 - 0.0393393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364513 - 0.0393393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.204i)T \) |
| 3 | \( 1 + (1.70 - 0.304i)T \) |
| 67 | \( 1 + (-7.02 - 4.20i)T \) |
| good | 5 | \( 1 + (2.29 - 1.04i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.382 - 0.401i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (3.69 + 0.352i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (1.78 + 0.343i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 2.49i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (1.63 - 1.71i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (3.38 - 1.35i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-1.80 - 1.04i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.19 + 6.21i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (-2.53 - 4.39i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.71 + 0.462i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (1.51 + 2.35i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (2.11 + 1.66i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-1.46 + 2.27i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.90 - 6.81i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 1.18i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-10.5 + 5.41i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 0.115i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (9.81 - 3.39i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-4.24 + 5.96i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-10.4 + 1.50i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.46 - 4.26i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.20380256623550876608039310950, −9.677217499124901584208860072519, −8.314476798196105781916040209010, −7.64782733777070571051266319871, −7.01971641890584967694778666150, −5.93441733109798673161011022195, −5.10781206228882815541217431510, −3.76083585993095312617402456589, −2.37492462398002679050289481778, −0.45877196357600657064482815743,
0.72898553558216643116917562263, 2.30858486680761512468415048377, 3.93489715989332840118312283399, 4.96528894342695578908008842439, 6.06034906256083960978724705651, 7.09037862636331127857819381024, 7.83002199797851420893621990172, 8.344019138241801456621507179295, 9.603274354172117605168520056467, 10.41773324514363754447364365143
