| L(s) = 1 | + (0.973 − 0.230i)2-s + (0.538 + 1.64i)3-s + (0.893 − 0.448i)4-s + (−0.200 − 0.269i)5-s + (0.904 + 1.47i)6-s + (1.19 − 0.784i)7-s + (0.766 − 0.642i)8-s + (−2.41 + 1.77i)9-s + (−0.256 − 0.215i)10-s + (−1.81 − 0.212i)11-s + (1.22 + 1.22i)12-s + (0.451 + 1.50i)13-s + (0.979 − 1.03i)14-s + (0.334 − 0.474i)15-s + (0.597 − 0.802i)16-s + (−0.187 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.688 − 0.163i)2-s + (0.311 + 0.950i)3-s + (0.446 − 0.224i)4-s + (−0.0895 − 0.120i)5-s + (0.369 + 0.603i)6-s + (0.450 − 0.296i)7-s + (0.270 − 0.227i)8-s + (−0.806 + 0.591i)9-s + (−0.0812 − 0.0681i)10-s + (−0.547 − 0.0640i)11-s + (0.352 + 0.354i)12-s + (0.125 + 0.418i)13-s + (0.261 − 0.277i)14-s + (0.0864 − 0.122i)15-s + (0.149 − 0.200i)16-s + (−0.0454 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69868 + 0.355286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69868 + 0.355286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.973 + 0.230i)T \) |
| 3 | \( 1 + (-0.538 - 1.64i)T \) |
| good | 5 | \( 1 + (0.200 + 0.269i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.19 + 0.784i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (1.81 + 0.212i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.451 - 1.50i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (0.187 - 1.06i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.689 + 3.90i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (7.14 + 4.70i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-0.506 - 0.537i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (0.386 - 6.64i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-8.37 + 3.04i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (3.61 + 0.855i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.214 - 0.497i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.159 - 2.73i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (4.39 - 7.61i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.28 - 0.384i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-12.4 - 6.24i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (2.32 - 2.45i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-4.50 - 3.78i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.351 - 0.295i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (5.46 - 1.29i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 2.64i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 9.85i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-5.90 + 7.93i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−13.04012230185583481442846825603, −11.88261456313816011340668645135, −10.86225651088275826342869187958, −10.20658653813635026531246491047, −8.874188592392374077483490838551, −7.85146060064725541312930581233, −6.26130786220955356695223386149, −4.88544609536686298375370376687, −4.11087052609390220557040529121, −2.56197349468275972828303079695,
2.06868121934322734864308495111, 3.55931688808551191912349073544, 5.33462011337427340837645004236, 6.30836994656133106543454241041, 7.66513710720660569144012053130, 8.178960836242593604790272304723, 9.753848482431447875594109638666, 11.25265394846439812725511100640, 11.95852651597599554734331972429, 12.97457841871104813548761402048
