| L(s) = 1 | + (1.98 + 0.278i)2-s + (−1.09 + 1.33i)3-s + (3.84 + 1.10i)4-s + (1.13 + 3.74i)5-s + (−2.54 + 2.34i)6-s + (−0.786 + 3.95i)7-s + (7.30 + 3.25i)8-s + (−0.585 − 2.94i)9-s + (1.20 + 7.72i)10-s + (11.9 + 1.18i)11-s + (−5.70 + 3.93i)12-s + (−22.2 − 6.76i)13-s + (−2.65 + 7.60i)14-s + (−6.25 − 2.59i)15-s + (13.5 + 8.47i)16-s + (11.0 + 26.6i)17-s + ⋯ |
| L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.366 + 0.446i)3-s + (0.961 + 0.275i)4-s + (0.227 + 0.748i)5-s + (−0.424 + 0.390i)6-s + (−0.112 + 0.564i)7-s + (0.913 + 0.406i)8-s + (−0.0650 − 0.326i)9-s + (0.120 + 0.772i)10-s + (1.08 + 0.107i)11-s + (−0.475 + 0.328i)12-s + (−1.71 − 0.520i)13-s + (−0.189 + 0.543i)14-s + (−0.417 − 0.172i)15-s + (0.848 + 0.529i)16-s + (0.650 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.83506 + 2.13679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83506 + 2.13679i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.98 - 0.278i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-1.13 - 3.74i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.786 - 3.95i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-11.9 - 1.18i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (22.2 + 6.76i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-11.0 - 26.6i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (2.03 - 1.08i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (21.6 + 14.4i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (3.18 - 0.313i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (22.7 - 22.7i)T - 961iT^{2} \) |
| 37 | \( 1 + (-53.8 - 28.7i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-2.79 - 1.86i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (8.83 + 10.7i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (31.4 + 75.9i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-81.9 - 8.06i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-3.61 - 11.9i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-63.0 + 76.7i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-23.8 - 19.5i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (45.9 + 9.14i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (98.2 - 19.5i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 + 124. i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (28.5 + 53.4i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (42.3 + 63.3i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-110. - 110. i)T + 9.40e3iT^{2} \) |
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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
−11.64596330123314594272726806180, −10.40387587260701941326826545186, −10.01272303216911984411624852519, −8.511223003166428897086830543116, −7.26913065524575697229610870163, −6.35545456976852084349057122486, −5.62295457980670744897495754630, −4.46671543632355933244319222505, −3.37218085218613778876531080189, −2.15213913118831205059998710326,
0.979452256952400176780420376121, 2.41090401507853725692760493039, 4.03409885167141668146872281190, 4.95063956769120846250443936998, 5.85774122028343216943104356589, 7.06913995302768951080400944881, 7.55257626539338405368555558295, 9.332453181888639474242704022238, 9.943498205611441572479496323314, 11.39419844546951459068347780802
