L(s) = 1 | + (−0.888 − 2.86i)3-s − 8.59·5-s + 10.9·7-s + (−7.41 + 5.09i)9-s − 2.75·11-s − 4.43i·13-s + (7.63 + 24.6i)15-s + 25.4i·17-s + 17.5i·19-s + (−9.71 − 31.3i)21-s − 17.5i·23-s + 48.8·25-s + (21.1 + 16.7i)27-s + 19.6·29-s − 2.58·31-s + ⋯ |
L(s) = 1 | + (−0.296 − 0.955i)3-s − 1.71·5-s + 1.56·7-s + (−0.824 + 0.565i)9-s − 0.250·11-s − 0.341i·13-s + (0.509 + 1.64i)15-s + 1.49i·17-s + 0.923i·19-s + (−0.462 − 1.49i)21-s − 0.762i·23-s + 1.95·25-s + (0.784 + 0.619i)27-s + 0.676·29-s − 0.0833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.223404814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.223404814\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.888 + 2.86i)T \) |
good | 5 | \( 1 + 8.59T + 25T^{2} \) |
| 7 | \( 1 - 10.9T + 49T^{2} \) |
| 11 | \( 1 + 2.75T + 121T^{2} \) |
| 13 | \( 1 + 4.43iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 17.5iT - 361T^{2} \) |
| 23 | \( 1 + 17.5iT - 529T^{2} \) |
| 29 | \( 1 - 19.6T + 841T^{2} \) |
| 31 | \( 1 + 2.58T + 961T^{2} \) |
| 37 | \( 1 + 7.73iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 17.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 50.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 32.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 48.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 22.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 27.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 97.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 59.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.1T + 9.40e3T^{2} \) |
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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
−10.66216591995789158896152849647, −8.587681636785636379478275008252, −8.123361587723768836351581987327, −7.75960159069001376521415038601, −6.76897006233751321357629651855, −5.57758035007589435176255152726, −4.59812493766798312321689990470, −3.64529325620887925618910514986, −2.07305912916284957338629832474, −0.800364356728166449762481897366,
0.69619718895081540557598009873, 2.80725670988240473242398526705, 3.98361752723338434901354403872, 4.72732702664300643046838622379, 5.23911731525791293671650770643, 6.93002066246200671108026519276, 7.71846746272599057841984307462, 8.468721999217045575930067057868, 9.209863555773137077918877878870, 10.41045934485207687849097143333