| L(s) = 1 | + (−0.978 + 0.207i)2-s + (−0.556 + 1.64i)3-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.203 − 1.72i)6-s + (−0.187 + 1.78i)7-s + (−0.809 + 0.587i)8-s + (−2.38 − 1.82i)9-s + 1.00·10-s + (0.353 − 3.29i)11-s + (0.158 + 1.72i)12-s + (−0.773 − 0.858i)13-s + (−0.187 − 1.78i)14-s + (0.885 − 1.48i)15-s + (0.669 − 0.743i)16-s + (−0.205 − 0.632i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (−0.321 + 0.946i)3-s + (0.456 − 0.203i)4-s + (−0.437 − 0.0929i)5-s + (0.0830 − 0.702i)6-s + (−0.0707 + 0.672i)7-s + (−0.286 + 0.207i)8-s + (−0.793 − 0.608i)9-s + 0.316·10-s + (0.106 − 0.994i)11-s + (0.0458 + 0.497i)12-s + (−0.214 − 0.238i)13-s + (−0.0500 − 0.475i)14-s + (0.228 − 0.384i)15-s + (0.167 − 0.185i)16-s + (−0.0498 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.807799 + 0.0778754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.807799 + 0.0778754i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.556 - 1.64i)T \) |
| 5 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 + (-0.353 + 3.29i)T \) |
| good | 7 | \( 1 + (0.187 - 1.78i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (0.773 + 0.858i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.205 + 0.632i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.00 + 4.36i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.235 + 0.408i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.468 - 4.45i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (1.37 + 1.52i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-6.06 - 4.40i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.01 + 9.62i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-1.58 - 2.74i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.03 + 2.68i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.19 + 6.76i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.12 + 1.83i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (3.18 - 3.54i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.791 - 2.43i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.59 - 6.97i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 0.266i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-5.84 + 6.49i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (-14.1 + 3.00i)T + (88.6 - 39.4i)T^{2} \) |
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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
−9.906151995485852544989503194664, −9.099644246297309136519296099131, −8.669597010089331703797713676682, −7.65759129803895808221934824117, −6.60974642103747154912992012105, −5.60448140895666753474517247105, −4.99082865070917485169272274607, −3.59667444081337075979158174952, −2.73748540020193367300483825439, −0.64394377523125295831752166398,
0.983031986654238089639581375833, 2.13491792896605732030810844464, 3.46045648782472202539367909993, 4.69923245512401444342929670535, 5.96051471086911604657835877013, 6.84713901465061973786296271265, 7.66337795042540566786316828793, 7.86710332747505639937790676654, 9.204090845781752868897801259692, 9.955686600232341763792470536390
