L(s) = 1 | + 3·3-s − 15.0i·5-s + (−17.2 + 6.82i)7-s + 9·9-s + 27.2i·11-s + 25.6i·13-s − 45.1i·15-s − 13.7i·17-s + 147.·19-s + (−51.6 + 20.4i)21-s − 87.4i·23-s − 101.·25-s + 27·27-s + 19.9·29-s + 24.8·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34i·5-s + (−0.929 + 0.368i)7-s + 0.333·9-s + 0.745i·11-s + 0.546i·13-s − 0.777i·15-s − 0.196i·17-s + 1.77·19-s + (−0.536 + 0.212i)21-s − 0.792i·23-s − 0.812·25-s + 0.192·27-s + 0.127·29-s + 0.143·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.256573498\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.256573498\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (17.2 - 6.82i)T \) |
good | 5 | \( 1 + 15.0iT - 125T^{2} \) |
| 11 | \( 1 - 27.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 25.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 13.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 19.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 216. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 527.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 77.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 671. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 706. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 254. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 38.8iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 33.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 188. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3iT - 9.12e5T^{2} \) |
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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
−9.079619821648511381102386651309, −8.540856233043013301114311670551, −7.48101891110173855045227673490, −6.77284236800461332082335231444, −5.58551240203725202684108895592, −4.85155661580762094448257979795, −3.92688500009959109580205101814, −2.87509941099186510148242725537, −1.73498209090107428651144428862, −0.57914957150779770040169379661,
0.955023807279980225806983643600, 2.57208428308574458765040961330, 3.28690017225958774357190133618, 3.75333771931073815985815253358, 5.40038280334925159744628452121, 6.18656372292944313663411226070, 7.20273225074807601557743121973, 7.45922500074986413501021693907, 8.634994819784321836134248755302, 9.489278800860534012855189289833