L(s) = 1 | − 1.84·2-s + (−1.39 − 1.02i)3-s + 1.39·4-s + (0.667 − 1.15i)5-s + (2.56 + 1.89i)6-s + 1.12·8-s + (0.880 + 2.86i)9-s + (−1.22 + 2.12i)10-s + (−0.756 − 1.31i)11-s + (−1.93 − 1.43i)12-s + (2.58 + 4.48i)13-s + (−2.11 + 0.923i)15-s − 4.84·16-s + (−0.774 + 1.34i)17-s + (−1.62 − 5.28i)18-s + (1.25 + 2.16i)19-s + ⋯ |
L(s) = 1 | − 1.30·2-s + (−0.804 − 0.594i)3-s + 0.695·4-s + (0.298 − 0.516i)5-s + (1.04 + 0.773i)6-s + 0.396·8-s + (0.293 + 0.955i)9-s + (−0.388 + 0.673i)10-s + (−0.228 − 0.395i)11-s + (−0.558 − 0.413i)12-s + (0.717 + 1.24i)13-s + (−0.547 + 0.238i)15-s − 1.21·16-s + (−0.187 + 0.325i)17-s + (−0.382 − 1.24i)18-s + (0.287 + 0.497i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.515667 - 0.217146i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.515667 - 0.217146i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.39 + 1.02i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 5 | \( 1 + (-0.667 + 1.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.756 + 1.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 4.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.68 + 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0309 - 0.0536i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + (2.80 - 4.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.703 + 1.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.09 + 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.81866498459777319582351199857, −10.20735942737701295104454032174, −8.922627912527083727206809710753, −8.581781776701065119319876529064, −7.35830747782597642568545626902, −6.60049782850361767073289305150, −5.48782731380195108105405114505, −4.31531864093039345581475280958, −2.01816875309205544487189948943, −0.860738025715233225981353269194,
0.989287337910654548576680380373, 2.98271630836785874010859626770, 4.56550204661676794427826111982, 5.65417577727565063870235479776, 6.74930008128797076166164546736, 7.64470817573030108043440872128, 8.733153672100647656500387155255, 9.650546842971226773045867162369, 10.22461564224855303268010190325, 10.93779960400879654437187340218