| L(s) = 1 | + (−1.07 + 0.541i)2-s + (−1.45 − 0.933i)3-s + (−0.325 + 0.437i)4-s + (1.12 − 3.76i)5-s + (2.07 + 0.215i)6-s + (0.736 − 1.70i)7-s + (0.533 − 3.02i)8-s + (1.25 + 2.72i)9-s + (0.823 + 4.66i)10-s + (−4.71 − 1.11i)11-s + (0.883 − 0.334i)12-s + (0.260 − 0.171i)13-s + (0.130 + 2.23i)14-s + (−5.15 + 4.44i)15-s + (0.749 + 2.50i)16-s + (3.17 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.762 + 0.382i)2-s + (−0.842 − 0.538i)3-s + (−0.162 + 0.218i)4-s + (0.504 − 1.68i)5-s + (0.848 + 0.0881i)6-s + (0.278 − 0.645i)7-s + (0.188 − 1.06i)8-s + (0.419 + 0.907i)9-s + (0.260 + 1.47i)10-s + (−1.42 − 0.337i)11-s + (0.254 − 0.0965i)12-s + (0.0721 − 0.0474i)13-s + (0.0348 + 0.598i)14-s + (−1.33 + 1.14i)15-s + (0.187 + 0.625i)16-s + (0.770 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.384036 - 0.293773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384036 - 0.293773i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.45 + 0.933i)T \) |
| good | 2 | \( 1 + (1.07 - 0.541i)T + (1.19 - 1.60i)T^{2} \) |
| 5 | \( 1 + (-1.12 + 3.76i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.736 + 1.70i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (4.71 + 1.11i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (-0.260 + 0.171i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 1.15i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.385i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.902 - 2.09i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.166 - 2.86i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-4.82 - 0.564i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (-8.69 + 7.29i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-1.29 - 0.648i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (3.07 - 3.25i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-3.90 + 0.456i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (0.812 + 1.40i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.72 - 2.30i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (0.456 + 0.613i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (0.0428 + 0.735i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.18 - 6.69i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.692 - 3.92i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-11.0 + 5.57i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-6.61 + 3.32i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (-0.943 + 5.35i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.32 + 4.41i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−13.61999546895060276271420816422, −13.03372490268812299672907949352, −12.26339857173207271472803159985, −10.66850492642041990428546318321, −9.561954698108976281084876709520, −8.232140172786644470472261133845, −7.57075855747668622687060769813, −5.76050321590421812166999945632, −4.64823519385393419668720624693, −0.938576788538077135468355605828,
2.64310319093005655054388844285, 5.13637949334715432021393239253, 6.20687801472466460407232737545, 7.80728317231192799424842715240, 9.572213194025027239633258420075, 10.27781499073454136073179087333, 10.91644430476537765070558206159, 11.91584772130611573843097789000, 13.66519880896931019773907955398, 14.84614718063466023549913110741
