| L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s + 2i·7-s + 8-s − 9-s − i·12-s + 2·13-s + 2i·14-s + 16-s − 18-s + 4·19-s + 2·21-s − 6i·23-s − i·24-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.755i·7-s + 0.353·8-s − 0.333·9-s − 0.288i·12-s + 0.554·13-s + 0.534i·14-s + 0.250·16-s − 0.235·18-s + 0.917·19-s + 0.436·21-s − 1.25i·23-s − 0.204i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.866837217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.866837217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.979173196591471656776370496740, −8.600393498931086055472204132468, −7.49103202799566771469869271032, −6.82648587556016349391726515741, −5.97177801198809475079072401890, −5.33661799282712875273111829402, −4.36646658667985856916874935541, −3.17590722233640677677728057480, −2.44006914185019566497453519431, −1.15271937090218071845493446153,
1.10724594754821778575580921776, 2.64494159055079278065636157421, 3.66575727459453219011272480671, 4.21228263715662767145466959834, 5.25797853329593195045243327853, 5.88496552626821830100945001989, 6.95929998615006268455890206570, 7.59855446566555361017515998893, 8.551567310141174737045651428910, 9.535250749395966050817422340301
