| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 2.82i·5-s + 2.22·7-s − 2.82i·8-s − 3.98·10-s + 4.52·13-s + 3.15i·14-s + 4.00·16-s − 14.7i·17-s − 11.0·19-s − 5.64i·20-s + 11.0i·23-s + 17.0·25-s + 6.39i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.564i·5-s + 0.318·7-s − 0.353i·8-s − 0.398·10-s + 0.347·13-s + 0.225i·14-s + 0.250·16-s − 0.866i·17-s − 0.583·19-s − 0.282i·20-s + 0.481i·23-s + 0.681·25-s + 0.246i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.559384358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559384358\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 25T^{2} \) |
| 7 | \( 1 - 2.22T + 49T^{2} \) |
| 13 | \( 1 - 4.52T + 169T^{2} \) |
| 17 | \( 1 + 14.7iT - 289T^{2} \) |
| 19 | \( 1 + 11.0T + 361T^{2} \) |
| 23 | \( 1 - 11.0iT - 529T^{2} \) |
| 29 | \( 1 - 22.6iT - 841T^{2} \) |
| 31 | \( 1 + 23.0T + 961T^{2} \) |
| 37 | \( 1 - 20.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 93.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 87.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 35.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 25.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 21.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 34.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−9.083773334849748290087761901090, −8.351456355094287582700993943671, −7.50877859454679713356811978500, −6.93500593219514292009329582669, −6.15290832365080363936213058689, −5.31088432690108724972203578345, −4.50425319981859874454148316287, −3.51319362636884330960681165897, −2.51868722174199322132153897770, −1.10385808252434066599348433807,
0.43515236429501363178043150308, 1.55872179016339895379993995106, 2.47184411753167306628654543176, 3.70939130453281320741096828029, 4.36317965630944093539249219357, 5.24807166297848235519089435479, 6.07146674074613313461488363849, 7.04434359719748780991151931228, 8.238821194966169337383524950749, 8.472686525265928127669948766790
