L(s) = 1 | + 2-s + (−0.420 + 1.68i)3-s + 4-s + 3.36i·5-s + (−0.420 + 1.68i)6-s + (−2.37 + 1.16i)7-s + 8-s + (−2.64 − 1.41i)9-s + 3.36i·10-s + (−0.420 + 1.68i)12-s + (2.79 − 2.27i)13-s + (−2.37 + 1.16i)14-s + (−5.64 − 1.41i)15-s + 16-s − 7.82·17-s + (−2.64 − 1.41i)18-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.242 + 0.970i)3-s + 0.5·4-s + 1.50i·5-s + (−0.171 + 0.685i)6-s + (−0.898 + 0.439i)7-s + 0.353·8-s + (−0.881 − 0.471i)9-s + 1.06i·10-s + (−0.121 + 0.485i)12-s + (0.775 − 0.631i)13-s + (−0.635 + 0.311i)14-s + (−1.45 − 0.365i)15-s + 0.250·16-s − 1.89·17-s + (−0.623 − 0.333i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542462 + 1.53868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542462 + 1.53868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.420 - 1.68i)T \) |
| 7 | \( 1 + (2.37 - 1.16i)T \) |
| 13 | \( 1 + (-2.79 + 2.27i)T \) |
good | 5 | \( 1 - 3.36iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 7.82T + 17T^{2} \) |
| 19 | \( 1 - 5.59T + 19T^{2} \) |
| 23 | \( 1 - 0.500iT - 23T^{2} \) |
| 29 | \( 1 - 5.15iT - 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 2.32iT - 37T^{2} \) |
| 41 | \( 1 - 9.87iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 4.33iT - 47T^{2} \) |
| 53 | \( 1 - 0.500iT - 53T^{2} \) |
| 59 | \( 1 + 2.16iT - 59T^{2} \) |
| 61 | \( 1 - 4.55iT - 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 0.708T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 3.14iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−11.13789302207165334563448184958, −10.45159959981278022106373045280, −9.662490373612771172808503593712, −8.649503609275701833415499193215, −7.17766116775721647687455154658, −6.35054301930417428649473449958, −5.73701279019015160388289336705, −4.41763164738188979053493111006, −3.25120527139660967557555024280, −2.80302607631594255486200183643,
0.76832849162960674276123471425, 2.19803920519537568686586169551, 3.83190489830823968882272076281, 4.82604497701024568513209364066, 5.88826194834147145149958870589, 6.63778599501639991485820283618, 7.57799996580627312432082274634, 8.703223446268029612659462845176, 9.335460336535678246637020115823, 10.76391276146304221658188930336