| L(s) = 1 | + (1.86 + 0.442i)2-s + (−1.20 + 1.24i)3-s + (1.50 + 0.757i)4-s + (0.470 − 0.632i)5-s + (−2.80 + 1.78i)6-s + (−0.995 − 0.654i)7-s + (−0.459 − 0.385i)8-s + (−0.0807 − 2.99i)9-s + (1.15 − 0.973i)10-s + (2.04 − 0.238i)11-s + (−2.76 + 0.956i)12-s + (−0.874 + 2.92i)13-s + (−1.57 − 1.66i)14-s + (0.216 + 1.34i)15-s + (−2.70 − 3.63i)16-s + (−0.788 − 4.47i)17-s + ⋯ |
| L(s) = 1 | + (1.32 + 0.313i)2-s + (−0.697 + 0.716i)3-s + (0.754 + 0.378i)4-s + (0.210 − 0.282i)5-s + (−1.14 + 0.728i)6-s + (−0.376 − 0.247i)7-s + (−0.162 − 0.136i)8-s + (−0.0269 − 0.999i)9-s + (0.366 − 0.307i)10-s + (0.615 − 0.0719i)11-s + (−0.797 + 0.276i)12-s + (−0.242 + 0.810i)13-s + (−0.419 − 0.444i)14-s + (0.0557 + 0.348i)15-s + (−0.675 − 0.907i)16-s + (−0.191 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31388 + 0.460640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31388 + 0.460640i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.20 - 1.24i)T \) |
| good | 2 | \( 1 + (-1.86 - 0.442i)T + (1.78 + 0.897i)T^{2} \) |
| 5 | \( 1 + (-0.470 + 0.632i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (0.995 + 0.654i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 0.238i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.874 - 2.92i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.788 + 4.47i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.838 - 4.75i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (5.50 - 3.62i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (-6.90 + 7.31i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.517 - 8.89i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (0.563 + 0.205i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (1.08 - 0.256i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (-2.96 + 6.87i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.0532 + 0.914i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-3.63 - 6.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.17 - 0.605i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (0.0194 - 0.00974i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (10.3 + 10.9i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-3.88 + 3.26i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.586 + 0.492i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.72 - 1.82i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (13.0 + 3.08i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (4.47 + 3.75i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.33 - 5.82i)T + (-27.8 + 92.9i)T^{2} \) |
| 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
−14.30804986264116689555262508853, −13.66536283963346778811139863015, −12.21957311663496739977618787242, −11.74664119495472372549052178912, −10.10248038655521112777447566864, −9.151171532163953311160584914713, −6.89170775592912289055535448120, −5.89197728592835188462045935119, −4.72690404258922029007854829979, −3.65558418676456775062389741285,
2.57677004160925351122849971394, 4.46458961341202810761207245714, 5.88433523207792799000759469518, 6.63280277861574622160831822809, 8.416635260051157062628922676200, 10.31718069334839428838897425651, 11.39938982192570580613485783278, 12.44170293372784239959878039053, 12.93614997671481556593808213550, 14.02124059489963179384395289936
