| 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 | \( 1 + (-5.90 - 7.93i)T + (-27.8 + 92.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97457841871104813548761402048, −11.95852651597599554734331972429, −11.25265394846439812725511100640, −9.753848482431447875594109638666, −8.178960836242593604790272304723, −7.66513710720660569144012053130, −6.30836994656133106543454241041, −5.33462011337427340837645004236, −3.55931688808551191912349073544, −2.06868121934322734864308495111,
2.56197349468275972828303079695, 4.11087052609390220557040529121, 4.88544609536686298375370376687, 6.26130786220955356695223386149, 7.85146060064725541312930581233, 8.874188592392374077483490838551, 10.20658653813635026531246491047, 10.86225651088275826342869187958, 11.88261456313816011340668645135, 13.04012230185583481442846825603
