| L(s) = 1 | + (−1.30 − 0.538i)2-s + (0.290 + 0.956i)3-s + (1.42 + 1.40i)4-s + (0.403 + 4.09i)5-s + (0.135 − 1.40i)6-s + (2.40 − 3.60i)7-s + (−1.10 − 2.60i)8-s + (−0.831 + 0.555i)9-s + (1.67 − 5.57i)10-s + (3.69 − 1.97i)11-s + (−0.934 + 1.76i)12-s + (2.07 + 0.204i)13-s + (−5.08 + 3.41i)14-s + (−3.80 + 1.57i)15-s + (0.0366 + 3.99i)16-s + (2.08 + 0.863i)17-s + ⋯ |
| L(s) = 1 | + (−0.924 − 0.380i)2-s + (0.167 + 0.552i)3-s + (0.710 + 0.703i)4-s + (0.180 + 1.83i)5-s + (0.0552 − 0.574i)6-s + (0.909 − 1.36i)7-s + (−0.389 − 0.921i)8-s + (−0.277 + 0.185i)9-s + (0.530 − 1.76i)10-s + (1.11 − 0.595i)11-s + (−0.269 + 0.510i)12-s + (0.574 + 0.0566i)13-s + (−1.35 + 0.912i)14-s + (−0.982 + 0.406i)15-s + (0.00915 + 0.999i)16-s + (0.505 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02916 + 0.466102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02916 + 0.466102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.538i)T \) |
| 3 | \( 1 + (-0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (-0.403 - 4.09i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (-2.40 + 3.60i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-3.69 + 1.97i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 0.204i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-2.08 - 0.863i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.762 - 0.625i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (1.23 - 6.19i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (2.11 - 3.95i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (0.316 - 0.316i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5.08 + 6.19i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (10.8 + 2.15i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (0.851 - 2.80i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-1.75 + 4.23i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.570 - 1.06i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (-3.98 + 0.392i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (0.499 - 0.151i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (2.49 - 0.757i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (4.03 + 2.69i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-3.48 - 5.21i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.944 - 2.27i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (10.3 + 12.6i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (3.30 + 16.5i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-3.83 + 3.83i)T - 97iT^{2} \) |
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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
−11.30493298448563342046511726356, −10.46660257825842153423145327736, −9.992689699659730629619776061633, −8.809067186271705427058726343105, −7.67446353918820237480647911666, −7.06828423139446610902278139787, −6.03337405914322027447449109575, −3.84834161796875362375089522822, −3.39181670480085254021770621205, −1.64976565311001266340760994993,
1.19587251069486967533558469086, 2.10653567102487411735531100310, 4.62151496689596351949286667914, 5.56507498240112441102843292519, 6.44177477392844184103073866249, 7.948521223406008824843972768265, 8.523577667038741800283631819648, 9.024180481371561279355707451184, 9.851189452846860304389172196564, 11.53161152356608461023042576740
