| L(s) = 1 | + (0.0928 − 0.346i)2-s + (−0.295 − 0.511i)3-s + (3.35 + 1.93i)4-s + (1.58 + 1.58i)5-s + (−0.204 + 0.0548i)6-s + (−0.0877 − 0.327i)7-s + (1.99 − 1.99i)8-s + (4.32 − 7.49i)9-s + (0.695 − 0.401i)10-s + (6.41 + 1.71i)11-s − 2.28i·12-s + (−9.96 + 8.34i)13-s − 0.121·14-s + (0.341 − 1.27i)15-s + (7.23 + 12.5i)16-s + (−25.1 − 14.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0464 − 0.173i)2-s + (−0.0983 − 0.170i)3-s + (0.838 + 0.483i)4-s + (0.316 + 0.316i)5-s + (−0.0341 + 0.00913i)6-s + (−0.0125 − 0.0467i)7-s + (0.249 − 0.249i)8-s + (0.480 − 0.832i)9-s + (0.0695 − 0.0401i)10-s + (0.583 + 0.156i)11-s − 0.190i·12-s + (−0.766 + 0.642i)13-s − 0.00868·14-s + (0.0227 − 0.0849i)15-s + (0.452 + 0.783i)16-s + (−1.48 − 0.854i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0493i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44070 - 0.0355515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44070 - 0.0355515i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.58 - 1.58i)T \) |
| 13 | \( 1 + (9.96 - 8.34i)T \) |
| good | 2 | \( 1 + (-0.0928 + 0.346i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (0.295 + 0.511i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (0.0877 + 0.327i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-6.41 - 1.71i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (25.1 + 14.5i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (27.1 - 7.26i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-10.5 + 6.10i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (19.0 + 33.0i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-27.5 - 27.5i)T + 961iT^{2} \) |
| 37 | \( 1 + (26.6 + 7.13i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (8.23 - 30.7i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (0.391 + 0.226i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-45.8 + 45.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 17.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (4.05 + 15.1i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-13.8 + 23.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (25.0 - 93.6i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (26.9 - 7.21i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (39.7 - 39.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 3.63T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-107. - 107. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (57.1 + 15.3i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-74.2 + 19.9i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−14.78463583393473850860122117918, −13.36538245021945655006773200078, −12.26242959083833174183494681971, −11.46224583496624294932843385082, −10.19744938535721542178585938052, −8.878571612979656362806518769868, −7.05931943616490371606111538914, −6.50717055306959156560191495999, −4.15641797247349840241160291879, −2.24812578397574785753676746515,
2.09942980093935593296543430865, 4.66618172275046871174149857158, 6.07383139984047393502336406564, 7.30414332472302421807084085148, 8.856519175901888827353594424181, 10.35650843840882658899806672890, 11.01897476973088208161479427578, 12.49665485936511518452596500570, 13.55551485821405653456120510121, 14.95017296308082734431044744086
