| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 4·5-s − 4.24i·7-s − 2.82i·8-s − 5.65i·10-s − 11·11-s − 16.9i·13-s + 6·14-s + 4.00·16-s − 4.24i·17-s − 21.2i·19-s + 8.00·20-s − 15.5i·22-s − 10·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.800·5-s − 0.606i·7-s − 0.353i·8-s − 0.565i·10-s − 11-s − 1.30i·13-s + 0.428·14-s + 0.250·16-s − 0.249i·17-s − 1.11i·19-s + 0.400·20-s − 0.707i·22-s − 0.434·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.398229 - 0.398229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398229 - 0.398229i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 + 4.24iT - 49T^{2} \) |
| 13 | \( 1 + 16.9iT - 169T^{2} \) |
| 17 | \( 1 + 4.24iT - 289T^{2} \) |
| 19 | \( 1 + 21.2iT - 361T^{2} \) |
| 23 | \( 1 + 10T + 529T^{2} \) |
| 29 | \( 1 - 21.2iT - 841T^{2} \) |
| 31 | \( 1 - 20T + 961T^{2} \) |
| 37 | \( 1 + 40T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40T + 2.20e3T^{2} \) |
| 53 | \( 1 - 20T + 2.80e3T^{2} \) |
| 59 | \( 1 + 70T + 3.48e3T^{2} \) |
| 61 | \( 1 - 84.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 40T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40T + 5.04e3T^{2} \) |
| 73 | \( 1 - 16.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 63.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 128T + 7.92e3T^{2} \) |
| 97 | \( 1 - 80T + 9.40e3T^{2} \) |
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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
−12.14181895043566037895302681774, −10.87708077048363484696480135973, −10.11748090742596758938117191618, −8.691874318457178456115317891831, −7.77332005523900710102317792229, −7.12086934000146092911825327914, −5.62228497910711361279506221564, −4.54239218501179213187657384212, −3.14871077904429995956794483335, −0.31501996981441443193841023420,
2.05483266237477368938917285982, 3.58669967855601349519979773242, 4.74531421309739844309891533381, 6.13031303085343006931059863572, 7.70101416945805916912693504619, 8.507556246551022978455901025529, 9.664221056321802154280237083840, 10.62677801843011722372040310289, 11.78164186015225353101385598194, 12.10993791139337200084156614163
