L(s) = 1 | + (−0.887 + 1.48i)3-s + (0.444 − 2.51i)5-s + (0.612 − 0.513i)7-s + (−1.42 − 2.63i)9-s + (−0.313 − 1.77i)11-s + (−2.44 + 0.888i)13-s + (3.35 + 2.89i)15-s + (3.13 − 5.43i)17-s + (−2.88 − 5.00i)19-s + (0.220 + 1.36i)21-s + (−1.80 − 1.51i)23-s + (−1.45 − 0.528i)25-s + (5.19 + 0.222i)27-s + (7.05 + 2.56i)29-s + (4.55 + 3.81i)31-s + ⋯ |
L(s) = 1 | + (−0.512 + 0.858i)3-s + (0.198 − 1.12i)5-s + (0.231 − 0.194i)7-s + (−0.475 − 0.879i)9-s + (−0.0946 − 0.536i)11-s + (−0.677 + 0.246i)13-s + (0.865 + 0.747i)15-s + (0.760 − 1.31i)17-s + (−0.662 − 1.14i)19-s + (0.0481 + 0.298i)21-s + (−0.377 − 0.316i)23-s + (−0.290 − 0.105i)25-s + (0.999 + 0.0428i)27-s + (1.31 + 0.476i)29-s + (0.817 + 0.685i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.917037 - 0.520131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.917037 - 0.520131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.887 - 1.48i)T \) |
good | 5 | \( 1 + (-0.444 + 2.51i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.612 + 0.513i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.313 + 1.77i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.44 - 0.888i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.88 + 5.00i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.80 + 1.51i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.05 - 2.56i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.55 - 3.81i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.0710 + 0.123i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.49 + 2.72i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.02 + 11.4i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.93 + 2.46i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + (0.688 - 3.90i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (10.0 - 8.39i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (13.7 - 5.01i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.27 + 5.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.483 + 0.837i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.693 - 0.252i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.18 - 2.25i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.90 + 5.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.957 + 5.43i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.92007988136387399634684493082, −10.12616457112992204565166718131, −9.138804371648405015150330829620, −8.654342729359469846488556320340, −7.27256275914191025041985928904, −6.03319951535323349345293665948, −4.91598191048454571011808386602, −4.57385277589465322684413731681, −2.92617490651327774122110299830, −0.73392344825847963496727386482,
1.77881888491871320625664133316, 2.93741327430357324466801374754, 4.59888337212364641629061476048, 6.03868608062283530249469810259, 6.39861653597530506188108739391, 7.72906528254185996646808497002, 8.077891666292537760639566015701, 9.849052807548470100252673273307, 10.44630532367766766994948188037, 11.29900740625858803940598129481