L(s) = 1 | + (0.866 − 0.5i)2-s + (−2.12 − 1.22i)3-s + (0.499 − 0.866i)4-s + (0.917 + 2.03i)5-s − 2.44·6-s − 0.999i·8-s + (1.49 + 2.59i)9-s + (1.81 + 1.30i)10-s + (2.44 − 4.24i)11-s + (−2.12 + 1.22i)12-s − 4.44i·13-s + (0.550 − 5.44i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (2.59 + 1.49i)18-s + (−0.775 − 1.34i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−1.22 − 0.707i)3-s + (0.249 − 0.433i)4-s + (0.410 + 0.911i)5-s − 0.999·6-s − 0.353i·8-s + (0.499 + 0.866i)9-s + (0.573 + 0.413i)10-s + (0.738 − 1.27i)11-s + (−0.612 + 0.353i)12-s − 1.23i·13-s + (0.142 − 1.40i)15-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.612 + 0.353i)18-s + (−0.177 − 0.308i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.759538 - 1.09065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759538 - 1.09065i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.917 - 2.03i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (2.12 + 1.22i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.44iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.775 + 1.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.51 + 1.44i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + (-4.44 + 7.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 - i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 + 0.898iT - 43T^{2} \) |
| 47 | \( 1 + (-7.70 + 4.44i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.43 - 5.44i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.775 + 1.34i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.77 - 3.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + (-2.51 - 1.44i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.44 - 5.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−11.01530788142293896167020100612, −10.30047960154137260042841952414, −9.034846383082327741135572735425, −7.57738417872254512045500015296, −6.63698769165705748563703554890, −5.96956063707369729132138902167, −5.36254887796711921358846715554, −3.73835134073787942977737456142, −2.52100673556246555625671644936, −0.796133082580594783924842149280,
1.78470312328218601087023553155, 4.09251718762061910817430981414, 4.60503778046362552846181793354, 5.46073139377254682183593793705, 6.37784439504044146613753448086, 7.20836613495209357893423167256, 8.767635917453732725866644501511, 9.496707243666267501077994359656, 10.40614896722209459306264233592, 11.47401053733710300620109270843