| L(s) = 1 | − 5.22i·3-s + 6.26·5-s + 2.64i·7-s − 18.2·9-s − 9.80i·11-s − 2.41·13-s − 32.7i·15-s + 6.89·17-s + 2.77i·19-s + 13.8·21-s − 42.8i·23-s + 14.2·25-s + 48.5i·27-s + 37.3·29-s − 7.16i·31-s + ⋯ |
| L(s) = 1 | − 1.74i·3-s + 1.25·5-s + 0.377i·7-s − 2.03·9-s − 0.891i·11-s − 0.185·13-s − 2.18i·15-s + 0.405·17-s + 0.146i·19-s + 0.658·21-s − 1.86i·23-s + 0.571·25-s + 1.79i·27-s + 1.28·29-s − 0.231i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.957847600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957847600\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 + 5.22iT - 9T^{2} \) |
| 5 | \( 1 - 6.26T + 25T^{2} \) |
| 11 | \( 1 + 9.80iT - 121T^{2} \) |
| 13 | \( 1 + 2.41T + 169T^{2} \) |
| 17 | \( 1 - 6.89T + 289T^{2} \) |
| 19 | \( 1 - 2.77iT - 361T^{2} \) |
| 23 | \( 1 + 42.8iT - 529T^{2} \) |
| 29 | \( 1 - 37.3T + 841T^{2} \) |
| 31 | \( 1 + 7.16iT - 961T^{2} \) |
| 37 | \( 1 + 0.202T + 1.36e3T^{2} \) |
| 41 | \( 1 + 63.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 35.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 37.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 104. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 31.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 69.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 5.11iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.4T + 9.40e3T^{2} \) |
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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
−8.351020227351506653708278589417, −8.145826884397396344588074425233, −6.81245629867482715265818902230, −6.41300892810952071549040377786, −5.79862484431663906463628321699, −4.92686271703898047923034424637, −3.11437742345688245032755242161, −2.36373360151601227456311689443, −1.54078686473574330440114678746, −0.48159348919847393421940213583,
1.57176568855703879662529472895, 2.82230718195276710402123564833, 3.71336325704055819207426947480, 4.69674443991293789333578423812, 5.24460537750176527746184470318, 6.00329372772262358069357638299, 7.03631794103423693827520666968, 8.105524610442052333586491656865, 9.223498480643452183085654643896, 9.497278945098542433447875857914
