L(s) = 1 | + 3.62i·2-s + (−1.98 − 2.24i)3-s − 9.11·4-s − 2.81i·5-s + (8.14 − 7.19i)6-s + 8.99·7-s − 18.5i·8-s + (−1.10 + 8.93i)9-s + 10.1·10-s − 19.0i·11-s + (18.0 + 20.4i)12-s + 8.47·13-s + 32.5i·14-s + (−6.32 + 5.58i)15-s + 30.5·16-s + 19.9i·17-s + ⋯ |
L(s) = 1 | + 1.81i·2-s + (−0.662 − 0.749i)3-s − 2.27·4-s − 0.562i·5-s + (1.35 − 1.19i)6-s + 1.28·7-s − 2.31i·8-s + (−0.123 + 0.992i)9-s + 1.01·10-s − 1.72i·11-s + (1.50 + 1.70i)12-s + 0.651·13-s + 2.32i·14-s + (−0.421 + 0.372i)15-s + 1.91·16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08256 + 0.409723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08256 + 0.409723i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.98 + 2.24i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 3.62iT - 4T^{2} \) |
| 5 | \( 1 + 2.81iT - 25T^{2} \) |
| 7 | \( 1 - 8.99T + 49T^{2} \) |
| 11 | \( 1 + 19.0iT - 121T^{2} \) |
| 13 | \( 1 - 8.47T + 169T^{2} \) |
| 17 | \( 1 - 19.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.3T + 361T^{2} \) |
| 23 | \( 1 + 10.3iT - 529T^{2} \) |
| 29 | \( 1 + 47.7iT - 841T^{2} \) |
| 31 | \( 1 - 4.05T + 961T^{2} \) |
| 37 | \( 1 + 13.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.46iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74.2iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 32.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 51.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 123.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 138.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 27.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 89.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 17.0T + 9.40e3T^{2} \) |
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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
−12.97933098427639563609329308884, −11.68398484726026863880142419882, −10.69494781256947025199233787909, −8.682244899461100370010383740275, −8.336374623078532211546375275701, −7.40879976094553884682562667197, −6.01746149899666844655128816930, −5.60246925380425717851985227967, −4.36514091712437406610423595112, −0.927585529922341034847976401248,
1.44896320266598951591141914237, 3.10640193621168284798465060449, 4.51214358556778799156511663961, 5.12279379075098095986690619705, 7.22972528460054163776443977003, 8.902401667476516055507428041402, 9.799868513570066424014090716402, 10.57420043033677905175045887461, 11.38123510579792995050983039331, 11.85630327039152000334254765052