L(s) = 1 | + 3i·3-s + 15.6i·5-s + 7·7-s − 9·9-s + 13.3i·11-s + 7.74i·13-s − 46.8·15-s − 25.3·17-s + 71.9i·19-s + 21i·21-s + 23.6·23-s − 118.·25-s − 27i·27-s − 9.49i·29-s − 74.7·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.39i·5-s + 0.377·7-s − 0.333·9-s + 0.366i·11-s + 0.165i·13-s − 0.806·15-s − 0.361·17-s + 0.869i·19-s + 0.218i·21-s + 0.214·23-s − 0.951·25-s − 0.192i·27-s − 0.0607i·29-s − 0.433·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9447636683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9447636683\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 - 15.6iT - 125T^{2} \) |
| 11 | \( 1 - 13.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 7.74iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 25.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 23.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 9.49iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 74.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 406.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 141. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 106. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 65.5iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 35.5iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 606. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 921.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 705.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 294.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 291. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 0.646T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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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
−10.01477214679026705317255829879, −9.006867840727638524750338536237, −8.101969030642893426160201417871, −7.23317764722763030807134483560, −6.53244012085382169780802096527, −5.62544240726412019249787572871, −4.59097693090137656276220144988, −3.65577565925420747340675346844, −2.81325610164003440963634247948, −1.72883195677385531507870849953,
0.22090101780678038292880617367, 1.15110464409834105358944627240, 2.14536574620839846543938455190, 3.48603519286071700621786792810, 4.72150707492854106033020333028, 5.21615893770273885246334462492, 6.22721994051193655593945167249, 7.18575610604368592562453687374, 8.059617063799959016839717398894, 8.744091567682190329952015186612