| L(s) = 1 | + (1.32 + 0.505i)2-s + (−0.281 − 1.70i)3-s + (1.48 + 1.33i)4-s + (−2.92 + 0.256i)5-s + (0.491 − 2.39i)6-s + (1.55 − 4.25i)7-s + (1.29 + 2.51i)8-s + (−2.84 + 0.963i)9-s + (−3.99 − 1.14i)10-s + (0.393 − 4.49i)11-s + (1.86 − 2.92i)12-s + (2.94 − 2.06i)13-s + (4.20 − 4.84i)14-s + (1.26 + 4.93i)15-s + (0.433 + 3.97i)16-s + (−1.67 + 2.90i)17-s + ⋯ |
| L(s) = 1 | + (0.933 + 0.357i)2-s + (−0.162 − 0.986i)3-s + (0.744 + 0.667i)4-s + (−1.30 + 0.114i)5-s + (0.200 − 0.979i)6-s + (0.585 − 1.60i)7-s + (0.456 + 0.889i)8-s + (−0.946 + 0.321i)9-s + (−1.26 − 0.361i)10-s + (0.118 − 1.35i)11-s + (0.537 − 0.843i)12-s + (0.816 − 0.571i)13-s + (1.12 − 1.29i)14-s + (0.326 + 1.27i)15-s + (0.108 + 0.994i)16-s + (−0.406 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54825 - 1.08084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54825 - 1.08084i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 - 0.505i)T \) |
| 3 | \( 1 + (0.281 + 1.70i)T \) |
| good | 5 | \( 1 + (2.92 - 0.256i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.55 + 4.25i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.393 + 4.49i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (-2.94 + 2.06i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.67 - 2.90i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.69 + 6.33i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.94 - 5.33i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.92 + 2.74i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (4.61 - 1.68i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.47 - 9.22i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.62 - 0.286i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0802 + 0.917i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.80 + 0.657i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.76 + 2.76i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.18 - 0.278i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-1.90 - 4.09i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (0.219 - 0.154i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-13.1 - 7.60i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.44 + 3.71i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.83 - 16.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.92 + 5.61i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (0.687 - 0.396i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.69 + 2.26i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.14994306848540674416481303862, −10.89833634722825306356887533245, −8.470625065819560437716343306275, −7.956678242421657190368081840485, −7.21004990233741029553688846918, −6.45734329717445859892196966183, −5.19693210738297816990977359832, −3.95236984661835134210358726759, −3.20378427862248769918803261483, −0.981836687538985137811820850252,
2.22466039383744286767189491846, 3.61986542333570206723906160596, 4.47691652754773133213522545473, 5.21389822752844649255814617470, 6.29002690832359948811978578867, 7.64898954606624777857415569952, 8.783168170244331443617926885815, 9.574476567056821629036819480995, 10.85380739500703979007709161820, 11.42431939957152760915168528544
