L(s) = 1 | + (0.246 + 2.98i)3-s − 6.63i·5-s + 0.578·7-s + (−8.87 + 1.47i)9-s + 8.68i·11-s + 17.9·13-s + (19.8 − 1.63i)15-s − 19.0i·17-s + 32.1·19-s + (0.142 + 1.72i)21-s − 20.4i·23-s − 19.0·25-s + (−6.59 − 26.1i)27-s + 22.0i·29-s + 26.2·31-s + ⋯ |
L(s) = 1 | + (0.0821 + 0.996i)3-s − 1.32i·5-s + 0.0825·7-s + (−0.986 + 0.163i)9-s + 0.789i·11-s + 1.37·13-s + (1.32 − 0.109i)15-s − 1.11i·17-s + 1.69·19-s + (0.00678 + 0.0823i)21-s − 0.890i·23-s − 0.761·25-s + (−0.244 − 0.969i)27-s + 0.759i·29-s + 0.846·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.82091 + 0.0749538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82091 + 0.0749538i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.246 - 2.98i)T \) |
good | 5 | \( 1 + 6.63iT - 25T^{2} \) |
| 7 | \( 1 - 0.578T + 49T^{2} \) |
| 11 | \( 1 - 8.68iT - 121T^{2} \) |
| 13 | \( 1 - 17.9T + 169T^{2} \) |
| 17 | \( 1 + 19.0iT - 289T^{2} \) |
| 19 | \( 1 - 32.1T + 361T^{2} \) |
| 23 | \( 1 + 20.4iT - 529T^{2} \) |
| 29 | \( 1 - 22.0iT - 841T^{2} \) |
| 31 | \( 1 - 26.2T + 961T^{2} \) |
| 37 | \( 1 - 53.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 30.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 88.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 63.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 33.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.60T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 48.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 93.3T + 9.40e3T^{2} \) |
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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
−11.20061092329133965436149638848, −9.998406078665805110679023147935, −9.300236578237207432860559159342, −8.632794861916531336186608050988, −7.64120792319083874926343585574, −6.05900692232077575161043915605, −4.97800081225600688712124014947, −4.41764685613205876467269186464, −3.03050922531264360173349890288, −1.03880484066540849245883298221,
1.22875853081949797982988819483, 2.81121875331713600259995411607, 3.66368705816402295080233917897, 5.84527256605935110303966202158, 6.24060933318572958926851053393, 7.40896722265235815331968313165, 8.067645744099440995520089890556, 9.156624487989610379511090514098, 10.45550602362944871373129619386, 11.25489753963352681016935165481