L(s) = 1 | + (−0.965 − 0.258i)2-s − i·3-s + (0.866 + 0.499i)4-s + (3.90 − 1.04i)5-s + (−0.258 + 0.965i)6-s + (−1.71 − 2.01i)7-s + (−0.707 − 0.707i)8-s − 9-s − 4.04·10-s + (−3.58 − 3.58i)11-s + (0.499 − 0.866i)12-s + (−2.15 − 2.88i)13-s + (1.13 + 2.39i)14-s + (−1.04 − 3.90i)15-s + (0.500 + 0.866i)16-s + (−2.48 + 4.29i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s − 0.577i·3-s + (0.433 + 0.249i)4-s + (1.74 − 0.468i)5-s + (−0.105 + 0.394i)6-s + (−0.648 − 0.761i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s − 1.27·10-s + (−1.08 − 1.08i)11-s + (0.144 − 0.249i)12-s + (−0.598 − 0.801i)13-s + (0.303 + 0.638i)14-s + (−0.270 − 1.00i)15-s + (0.125 + 0.216i)16-s + (−0.601 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.478397 - 0.966557i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.478397 - 0.966557i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (1.71 + 2.01i)T \) |
| 13 | \( 1 + (2.15 + 2.88i)T \) |
good | 5 | \( 1 + (-3.90 + 1.04i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (3.58 + 3.58i)T + 11iT^{2} \) |
| 17 | \( 1 + (2.48 - 4.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 1.02i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.70 + 2.71i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.24 - 7.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.06 + 3.97i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.202 - 0.754i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (3.46 - 0.927i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.45 + 4.88i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.761 + 2.84i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.08 - 3.61i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.242 - 0.903i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 12.1iT - 61T^{2} \) |
| 67 | \( 1 + (-6.50 + 6.50i)T - 67iT^{2} \) |
| 71 | \( 1 + (4.28 + 1.14i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-12.8 - 3.43i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 2.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.02 - 1.02i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.18 - 1.65i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.62 + 9.79i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.59661530567058771973855048248, −9.615904428052340455435046635967, −8.824156579762811028448012151136, −7.939189411687148569372057323423, −6.84177329433842802400352694229, −5.99381998423224723177235525192, −5.20663526349887678660647364949, −3.19191211632912126219469615602, −2.13541887598664600908119463117, −0.73586734529821199158976607717,
2.19773595195751152545412144984, 2.72759871743469685894322362587, 4.90737151258642547460747241004, 5.60425876048408855071127828698, 6.63047467072033467334311387053, 7.36462079638387283944740059918, 8.960818361829700080481312926101, 9.558636003263084979738803902481, 9.834843560473859790469751472633, 10.75768091025179779951156774923