| L(s) = 1 | + (−1.02 − 0.970i)2-s + (−2.72 + 1.25i)3-s + (0.116 + 1.99i)4-s + (0.295 − 2.52i)5-s + (4.01 + 1.35i)6-s + (−6.23 + 3.13i)7-s + (1.81 − 2.16i)8-s + (5.87 − 6.82i)9-s + (−2.75 + 2.31i)10-s + (10.3 + 4.47i)11-s + (−2.81 − 5.29i)12-s + (5.22 − 1.23i)13-s + (9.45 + 2.83i)14-s + (2.35 + 7.25i)15-s + (−3.97 + 0.464i)16-s + (26.6 − 4.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 − 0.485i)2-s + (−0.908 + 0.416i)3-s + (0.0290 + 0.499i)4-s + (0.0590 − 0.504i)5-s + (0.669 + 0.226i)6-s + (−0.891 + 0.447i)7-s + (0.227 − 0.270i)8-s + (0.652 − 0.757i)9-s + (−0.275 + 0.231i)10-s + (0.943 + 0.406i)11-s + (−0.234 − 0.441i)12-s + (0.401 − 0.0951i)13-s + (0.675 + 0.202i)14-s + (0.156 + 0.483i)15-s + (−0.248 + 0.0290i)16-s + (1.56 − 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.796341 - 0.232417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.796341 - 0.232417i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.970i)T \) |
| 3 | \( 1 + (2.72 - 1.25i)T \) |
| good | 5 | \( 1 + (-0.295 + 2.52i)T + (-24.3 - 5.76i)T^{2} \) |
| 7 | \( 1 + (6.23 - 3.13i)T + (29.2 - 39.3i)T^{2} \) |
| 11 | \( 1 + (-10.3 - 4.47i)T + (83.0 + 88.0i)T^{2} \) |
| 13 | \( 1 + (-5.22 + 1.23i)T + (151. - 75.8i)T^{2} \) |
| 17 | \( 1 + (-26.6 + 4.70i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (-4.51 + 25.5i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (-0.334 + 0.665i)T + (-315. - 424. i)T^{2} \) |
| 29 | \( 1 + (-29.3 + 8.78i)T + (702. - 462. i)T^{2} \) |
| 31 | \( 1 + (2.15 - 1.41i)T + (380. - 882. i)T^{2} \) |
| 37 | \( 1 + (55.2 + 20.1i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (-18.9 + 17.9i)T + (97.7 - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-39.2 - 52.7i)T + (-530. + 1.77e3i)T^{2} \) |
| 47 | \( 1 + (26.6 - 40.5i)T + (-874. - 2.02e3i)T^{2} \) |
| 53 | \( 1 + (-16.6 + 9.60i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-87.7 + 37.8i)T + (2.38e3 - 2.53e3i)T^{2} \) |
| 61 | \( 1 + (2.75 - 47.2i)T + (-3.69e3 - 431. i)T^{2} \) |
| 67 | \( 1 + (0.127 - 0.427i)T + (-3.75e3 - 2.46e3i)T^{2} \) |
| 71 | \( 1 + (42.0 + 50.1i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (75.1 + 63.0i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-38.4 + 40.7i)T + (-362. - 6.23e3i)T^{2} \) |
| 83 | \( 1 + (42.1 + 39.7i)T + (400. + 6.87e3i)T^{2} \) |
| 89 | \( 1 + (-59.8 + 71.3i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-75.6 + 8.84i)T + (9.15e3 - 2.16e3i)T^{2} \) |
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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
−12.29407152767711968952010113366, −11.62506706614408443401209014985, −10.45888382712396728013365725947, −9.527722256679829270032518151533, −8.919398019410691222803649039047, −7.18181860018246435375463138823, −6.07006711118508693028197840163, −4.75405231060485100919410470419, −3.28627391828181205965318133964, −0.934332803638337212548819074537,
1.12709026378419418809568402315, 3.63641736936238356616418382310, 5.56753659030735445701505624689, 6.45611405044514091164288290370, 7.19005335371051660311109474660, 8.459991287240201772046653446231, 9.996390799854754221667525786587, 10.44090996944637203020164883040, 11.72541082164893256765635682323, 12.55215845655239425987997406858
