| L(s) = 1 | + 0.828i·3-s − 3.65·5-s + 9.65i·7-s + 8.31·9-s + 18.4i·11-s − 11.6·13-s − 3.02i·15-s + 9.31·17-s − 15.1i·19-s − 7.99·21-s + 22.3i·23-s − 11.6·25-s + 14.3i·27-s + 28.3·29-s − 45.2i·31-s + ⋯ |
| L(s) = 1 | + 0.276i·3-s − 0.731·5-s + 1.37i·7-s + 0.923·9-s + 1.68i·11-s − 0.896·13-s − 0.201i·15-s + 0.547·17-s − 0.798i·19-s − 0.380·21-s + 0.971i·23-s − 0.465·25-s + 0.531i·27-s + 0.977·29-s − 1.45i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.819927 + 0.819927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819927 + 0.819927i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 0.828iT - 9T^{2} \) |
| 5 | \( 1 + 3.65T + 25T^{2} \) |
| 7 | \( 1 - 9.65iT - 49T^{2} \) |
| 11 | \( 1 - 18.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 - 9.31T + 289T^{2} \) |
| 19 | \( 1 + 15.1iT - 361T^{2} \) |
| 23 | \( 1 - 22.3iT - 529T^{2} \) |
| 29 | \( 1 - 28.3T + 841T^{2} \) |
| 31 | \( 1 + 45.2iT - 961T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 46.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 9.11iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 113.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 45.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 16.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 70.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 94.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 110.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25.4T + 9.40e3T^{2} \) |
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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
−13.07248214761170527710994204980, −12.21340043375954422548090115623, −11.62160579652216746664337767405, −9.960572664174744278269864086772, −9.411368563485731192469475419993, −7.904628345107630551411943441739, −6.99701013448217858902334694766, −5.27480011879201854044680317513, −4.20845557365760383641685889693, −2.31233399639340128453410015207,
0.846174030510180526995593938771, 3.42564652837675398985673233387, 4.57826025008553777757176962047, 6.40447539936554957921610267832, 7.51892916851241766223056277987, 8.255719499900388781735958368450, 9.956698144552444040160728504526, 10.73133405526317766593530517480, 11.86625964854225562436148975455, 12.85810608843695616041409454678
