| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 8.92·5-s + 10.9i·7-s + 2.82i·8-s − 12.6i·10-s + (−1.19 + 10.9i)11-s + 8.48i·13-s + 15.4·14-s + 4.00·16-s − 20.2i·17-s − 6.03i·19-s − 17.8·20-s + (15.4 + 1.69i)22-s + 18.3·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.78·5-s + 1.56i·7-s + 0.353i·8-s − 1.26i·10-s + (−0.108 + 0.994i)11-s + 0.652i·13-s + 1.10·14-s + 0.250·16-s − 1.19i·17-s − 0.317i·19-s − 0.892·20-s + (0.702 + 0.0768i)22-s + 0.799·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.86623 - 0.101770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86623 - 0.101770i\) |
| \(L(2)\) |
|
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 | \( 1 + (1.19 - 10.9i)T \) |
| good | 5 | \( 1 - 8.92T + 25T^{2} \) |
| 7 | \( 1 - 10.9iT - 49T^{2} \) |
| 13 | \( 1 - 8.48iT - 169T^{2} \) |
| 17 | \( 1 + 20.2iT - 289T^{2} \) |
| 19 | \( 1 + 6.03iT - 361T^{2} \) |
| 23 | \( 1 - 18.3T + 529T^{2} \) |
| 29 | \( 1 + 38.5iT - 841T^{2} \) |
| 31 | \( 1 + 14.1T + 961T^{2} \) |
| 37 | \( 1 + 31.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 30.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 25.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 35.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 7.17iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 98.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 94.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 4.11iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 43.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50T + 7.92e3T^{2} \) |
| 97 | \( 1 - 34.7T + 9.40e3T^{2} \) |
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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
−12.26248904406634847360954742930, −11.35726678034958023672638999980, −10.01615900665537926599578009778, −9.411466738852451234856746850727, −8.797220271649461364836151011899, −6.89246355661039910327861354416, −5.66391147786913792279911749878, −4.90427724929930242926838130290, −2.63450390016057386326073984080, −1.93308810168003225804639357930,
1.26305961927390477833925687188, 3.43274673994640464524840371157, 5.08641682699902306000811636230, 6.05383108273242444638791339894, 6.92575634096346305859090333455, 8.196817463185993703732464147615, 9.276751203101525691714137738329, 10.40396756553768829928592187944, 10.72590999072283249116536677951, 12.90468532132678334782906539369
