L(s) = 1 | + i·2-s + (0.420 + 1.68i)3-s − 4-s + (−1.68 + 0.420i)6-s + (2.57 + 0.595i)7-s − i·8-s + (−2.64 + 1.41i)9-s + 2.82i·11-s + (−0.420 − 1.68i)12-s − 3.36i·13-s + (−0.595 + 2.57i)14-s + 16-s − 4.75·17-s + (−1.41 − 2.64i)18-s + 5.59i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.242 + 0.970i)3-s − 0.5·4-s + (−0.685 + 0.171i)6-s + (0.974 + 0.224i)7-s − 0.353i·8-s + (−0.881 + 0.471i)9-s + 0.852i·11-s + (−0.121 − 0.485i)12-s − 0.931i·13-s + (−0.159 + 0.688i)14-s + 0.250·16-s − 1.15·17-s + (−0.333 − 0.623i)18-s + 1.28i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.415892973\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.415892973\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.420 - 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.57 - 0.595i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 3.36iT - 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 7.29iT - 23T^{2} \) |
| 29 | \( 1 - 0.500iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 53 | \( 1 + 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.81iT - 71T^{2} \) |
| 73 | \( 1 + 5.53iT - 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−10.15556388185952603385098138625, −9.466576925736303122382590452915, −8.546621355272785388196018672594, −8.000129521333982309386824198470, −7.13771507882223614661396077890, −5.81164247494007044128868034996, −5.17278501362149277408084071036, −4.38908475010394838988365631512, −3.40927230573629820008962725811, −1.91410671156693025954759156710,
0.61244916320849836865222375456, 1.93034533380014404218023767797, 2.73062089176082522804287425580, 4.11582508808924550008013510743, 4.96587008156821481718892729621, 6.24985128782271044734513990600, 6.98371238878246537904849381538, 8.021014343253394407425350630419, 8.743640160361461164968871725390, 9.210295391073221953323945127112