| L(s) = 1 | + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + 2.44i·6-s + (5.10 + 4.79i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (0.919 − 1.59i)11-s + (−2.99 + 1.73i)12-s + 5.40i·13-s + (−2.26 + 9.63i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (8.71 + 5.02i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.408i·6-s + (0.728 + 0.684i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (0.0835 − 0.144i)11-s + (−0.249 + 0.144i)12-s + 0.415i·13-s + (−0.161 + 0.688i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.512 + 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.72298 + 1.55294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72298 + 1.55294i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (-5.10 - 4.79i)T \) |
| good | 11 | \( 1 + (-0.919 + 1.59i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 5.40iT - 169T^{2} \) |
| 17 | \( 1 + (-8.71 - 5.02i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (7.96 - 4.59i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-0.460 - 0.797i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 12.5T + 841T^{2} \) |
| 31 | \( 1 + (36.1 + 20.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (3.64 + 6.30i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.12T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-29.4 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-52.0 + 90.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-12.5 - 7.26i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.5 + 11.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.46 + 12.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 17.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (107. + 62.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-40.4 - 70.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 154. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-58.9 + 34.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 88.1iT - 9.40e3T^{2} \) |
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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
−12.52030718579872585267073101364, −11.55847843963443843462378811560, −10.29740872466506484862339480236, −9.089396780639354044490515236916, −8.466613061485183875347145564546, −7.37740350400127627603624829971, −5.97707465412467643458967092126, −5.05206445565510467516714231971, −3.79882969927901047404357595070, −2.10498006632038874405151502940,
1.32873314219706409686147245452, 2.78725387599438033204172849071, 4.14854185842066092438285550327, 5.40679356147192571372529421036, 6.84143380173565019352715442682, 7.905606467825844605272081804082, 9.053651492908161167561698530032, 10.16286227359566795383774764080, 10.90278689580544019622415408684, 11.97763924417380998063981208352
