| L(s) = 1 | − 3i·3-s − 6i·5-s + 16·7-s − 9·9-s − 12i·11-s + 38i·13-s − 18·15-s − 126·17-s + 20i·19-s − 48i·21-s − 168·23-s + 89·25-s + 27i·27-s + 30i·29-s − 88·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.536i·5-s + 0.863·7-s − 0.333·9-s − 0.328i·11-s + 0.810i·13-s − 0.309·15-s − 1.79·17-s + 0.241i·19-s − 0.498i·21-s − 1.52·23-s + 0.711·25-s + 0.192i·27-s + 0.192i·29-s − 0.509·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 6iT - 125T^{2} \) |
| 7 | \( 1 - 16T + 343T^{2} \) |
| 11 | \( 1 + 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 126T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 168T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 88T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 42T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 96T + 1.03e5T^{2} \) |
| 53 | \( 1 + 198iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 660iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 538iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 884iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 792T + 3.57e5T^{2} \) |
| 73 | \( 1 + 218T + 3.89e5T^{2} \) |
| 79 | \( 1 + 520T + 4.93e5T^{2} \) |
| 83 | \( 1 + 492iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−9.045355283944169556423918986879, −8.635527349977539747011059829983, −7.71842667830765278883420758693, −6.79607051684719350040688854980, −5.89490957806419267866056687711, −4.80812291537779864572969360517, −4.00223952033311384578301692134, −2.32314795007014980898382008473, −1.47882309135144815357803456528, 0,
1.86583473427032091233652659999, 2.96440634633205244086595789393, 4.22693213939216774204977394023, 4.92668916845336885518751298676, 6.04759582014127750420073460243, 6.98699398808047146551979605127, 8.022020101694580846596646665026, 8.701533324920314270409308117076, 9.728764591749159091974519878510
