L(s) = 1 | + 1.41i·2-s + (0.707 + 0.292i)3-s − 2.00·4-s + (−0.414 + 1.00i)6-s + (−1 + i)7-s − 2.82i·8-s + (−1.70 − 1.70i)9-s + (−4.12 + 1.70i)11-s + (−1.41 − 0.585i)12-s + (−0.292 + 0.707i)13-s + (−1.41 − 1.41i)14-s + 4.00·16-s + 2.82i·17-s + (2.41 − 2.41i)18-s + (1.53 − 3.70i)19-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (0.408 + 0.169i)3-s − 1.00·4-s + (−0.169 + 0.408i)6-s + (−0.377 + 0.377i)7-s − 1.00i·8-s + (−0.569 − 0.569i)9-s + (−1.24 + 0.514i)11-s + (−0.408 − 0.169i)12-s + (−0.0812 + 0.196i)13-s + (−0.377 − 0.377i)14-s + 1.00·16-s + 0.685i·17-s + (0.569 − 0.569i)18-s + (0.352 − 0.850i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 - 0.292i)T + (2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + (4.12 - 1.70i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.292 - 0.707i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + (-1.53 + 3.70i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (5.82 + 5.82i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.12 + 1.29i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (0.292 + 0.707i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (0.171 + 0.171i)T + 41iT^{2} \) |
| 43 | \( 1 + (4.70 - 1.94i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 0.343iT - 47T^{2} \) |
| 53 | \( 1 + (-1.12 + 0.464i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.87 + 4.53i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 0.707i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-5.53 - 2.29i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (5.82 - 5.82i)T - 71iT^{2} \) |
| 73 | \( 1 + (7 + 7i)T + 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (1.87 - 4.53i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-8.65 + 8.65i)T - 89iT^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−9.770989852915791038973192794945, −9.026959375219524535402227285098, −8.292609621317742343109269376971, −7.53343111764430450896276565075, −6.47842740048033495269320342687, −5.74337484532709293090788605601, −4.75666008916035581234821039073, −3.67755908380458093349812821284, −2.48302901028086144495066347102, 0,
1.87417386993632887036259059602, 2.97556159879995253415990871864, 3.71471335167131132203540339668, 5.16683603362759257068352797762, 5.73555343818933855075921472411, 7.49522591645279530623130051672, 8.004550216472812858035360192627, 8.912186880040474685730745329412, 9.872646605310008168985372169941