L(s) = 1 | + (1.20 + 0.212i)2-s + (−0.477 − 0.173i)4-s + (−2.13 + 0.670i)5-s + (3.28 − 1.89i)7-s + (−2.65 − 1.53i)8-s + (−2.70 + 0.354i)10-s + (−0.618 + 1.07i)11-s + (−2.22 − 2.64i)13-s + (4.35 − 1.58i)14-s + (−2.08 − 1.75i)16-s + (−2.96 − 0.522i)17-s + (−4.28 − 0.793i)19-s + (1.13 + 0.0503i)20-s + (−0.971 + 1.15i)22-s + (2.10 − 5.77i)23-s + ⋯ |
L(s) = 1 | + (0.850 + 0.149i)2-s + (−0.238 − 0.0868i)4-s + (−0.953 + 0.300i)5-s + (1.24 − 0.716i)7-s + (−0.937 − 0.541i)8-s + (−0.856 + 0.112i)10-s + (−0.186 + 0.323i)11-s + (−0.616 − 0.734i)13-s + (1.16 − 0.423i)14-s + (−0.522 − 0.438i)16-s + (−0.718 − 0.126i)17-s + (−0.983 − 0.181i)19-s + (0.253 + 0.0112i)20-s + (−0.207 + 0.246i)22-s + (0.438 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.605786 - 0.931614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.605786 - 0.931614i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.13 - 0.670i)T \) |
| 19 | \( 1 + (4.28 + 0.793i)T \) |
good | 2 | \( 1 + (-1.20 - 0.212i)T + (1.87 + 0.684i)T^{2} \) |
| 7 | \( 1 + (-3.28 + 1.89i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.618 - 1.07i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.22 + 2.64i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.96 + 0.522i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 5.77i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.744 + 4.22i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.13iT - 37T^{2} \) |
| 41 | \( 1 + (4.08 + 3.42i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.12 + 8.57i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-7.19 + 1.26i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.13 - 3.13i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.141 - 0.804i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.01 + 2.18i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.995 + 0.175i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 4.67i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.11 - 8.47i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.06 - 0.889i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.18 - 1.26i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.06 + 1.73i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.01 - 0.531i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.19537019560436373465926272362, −8.845908446899901294510750956561, −8.129487716180056163396303741138, −7.29142976677848410016605799935, −6.44656617192425844725733899736, −5.08845074097227650788140481096, −4.54351465768492678793117166434, −3.84693375570630325147158611481, −2.48428609994341642726631921018, −0.40325808669701484099962329664,
1.92795220766973573429303228591, 3.26704346099029811658951725215, 4.34058560710998127320136521832, 4.88956357194032307379129332491, 5.70179079345863738868613797657, 7.04649599737568898915888492231, 8.056468308449783112503727274259, 8.707731460630141540828997754302, 9.281894389917805198057877788019, 10.97282117976148350446362762216