| L(s) = 1 | + (1.92 + 3.99i)2-s + (−6.56 − 9.62i)3-s + (−2.29 + 2.87i)4-s + (1.51 − 1.63i)5-s + (25.8 − 44.7i)6-s + (69.4 − 40.0i)7-s + (53.2 + 12.1i)8-s + (−19.9 + 50.8i)9-s + (9.43 + 2.91i)10-s + (−98.8 − 123. i)11-s + (42.7 + 3.20i)12-s + (−23.9 + 7.38i)13-s + (293. + 200. i)14-s + (−25.6 − 3.86i)15-s + (67.0 + 293. i)16-s + (−210. + 195. i)17-s + ⋯ |
| L(s) = 1 | + (0.481 + 0.999i)2-s + (−0.728 − 1.06i)3-s + (−0.143 + 0.179i)4-s + (0.0605 − 0.0652i)5-s + (0.717 − 1.24i)6-s + (1.41 − 0.818i)7-s + (0.832 + 0.190i)8-s + (−0.246 + 0.627i)9-s + (0.0943 + 0.0291i)10-s + (−0.817 − 1.02i)11-s + (0.296 + 0.0222i)12-s + (−0.141 + 0.0436i)13-s + (1.49 + 1.02i)14-s + (−0.113 − 0.0171i)15-s + (0.261 + 1.14i)16-s + (−0.728 + 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.72758 - 0.330721i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72758 - 0.330721i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.80e3 - 399. i)T \) |
| good | 2 | \( 1 + (-1.92 - 3.99i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (6.56 + 9.62i)T + (-29.5 + 75.4i)T^{2} \) |
| 5 | \( 1 + (-1.51 + 1.63i)T + (-46.7 - 623. i)T^{2} \) |
| 7 | \( 1 + (-69.4 + 40.0i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (98.8 + 123. i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (23.9 - 7.38i)T + (2.35e4 - 1.60e4i)T^{2} \) |
| 17 | \( 1 + (210. - 195. i)T + (6.24e3 - 8.32e4i)T^{2} \) |
| 19 | \( 1 + (-387. + 152. i)T + (9.55e4 - 8.86e4i)T^{2} \) |
| 23 | \( 1 + (139. - 20.9i)T + (2.67e5 - 8.24e4i)T^{2} \) |
| 29 | \( 1 + (-475. + 697. i)T + (-2.58e5 - 6.58e5i)T^{2} \) |
| 31 | \( 1 + (76.6 - 1.02e3i)T + (-9.13e5 - 1.37e5i)T^{2} \) |
| 37 | \( 1 + (-1.58e3 - 916. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.46e3 - 706. i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (281. - 352. i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (2.54e3 + 785. i)T + (6.51e6 + 4.44e6i)T^{2} \) |
| 59 | \( 1 + (-1.02e3 - 4.48e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (3.18e3 - 238. i)T + (1.36e7 - 2.06e6i)T^{2} \) |
| 67 | \( 1 + (-816. - 2.07e3i)T + (-1.47e7 + 1.37e7i)T^{2} \) |
| 71 | \( 1 + (195. - 1.29e3i)T + (-2.42e7 - 7.49e6i)T^{2} \) |
| 73 | \( 1 + (1.41e3 + 4.59e3i)T + (-2.34e7 + 1.59e7i)T^{2} \) |
| 79 | \( 1 + (3.45e3 + 5.98e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-8.32e3 + 5.67e3i)T + (1.73e7 - 4.41e7i)T^{2} \) |
| 89 | \( 1 + (-5.30e3 - 7.78e3i)T + (-2.29e7 + 5.84e7i)T^{2} \) |
| 97 | \( 1 + (7.67e3 + 9.62e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.07682008989953228452086504929, −13.83624227122302085194807698489, −13.26154824338204200974529448664, −11.54167166103335735899184118469, −10.74676828120622875524010820873, −8.102161145392248554477648420954, −7.28928283754221783131935956255, −6.00775229409046306315326221198, −4.84035532348860543878833284204, −1.27080532235973079614574287132,
2.28010389717264442484531650129, 4.47373162850767280546281365436, 5.23046677603477910878177783333, 7.75920184846724645227830865137, 9.698414589043306087547593662132, 10.78766733002892961139725490415, 11.56210945784968317916669285338, 12.46381054681072290716058546729, 14.07277150716978119726422729882, 15.34468233124954491806317340117
