| L(s) = 1 | − i·3-s − 2.49i·5-s + 0.917·7-s − 9-s + 3.69i·11-s + 5.81i·13-s − 2.49·15-s − 0.867·17-s − 6.52i·19-s − 0.917i·21-s + 4·23-s − 1.23·25-s + i·27-s − 7.72i·29-s + 2.14·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.11i·5-s + 0.346·7-s − 0.333·9-s + 1.11i·11-s + 1.61i·13-s − 0.644·15-s − 0.210·17-s − 1.49i·19-s − 0.200i·21-s + 0.834·23-s − 0.246·25-s + 0.192i·27-s − 1.43i·29-s + 0.385·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794109282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794109282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.49iT - 5T^{2} \) |
| 7 | \( 1 - 0.917T + 7T^{2} \) |
| 11 | \( 1 - 3.69iT - 11T^{2} \) |
| 13 | \( 1 - 5.81iT - 13T^{2} \) |
| 17 | \( 1 + 0.867T + 17T^{2} \) |
| 19 | \( 1 + 6.52iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.72iT - 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.47iT - 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 9.58iT - 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.39iT - 53T^{2} \) |
| 59 | \( 1 + 12.7iT - 59T^{2} \) |
| 61 | \( 1 - 0.0231iT - 61T^{2} \) |
| 67 | \( 1 + 5.32iT - 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 + 1.96iT - 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−8.524980222034830192657509063048, −7.81538644052943506799707281469, −6.90397246805566568625566765068, −6.53730223321632920860341921801, −5.21690359433966120043096086459, −4.70939549573465247528047838411, −4.06157903652546826653329820359, −2.47277603354276802653660681244, −1.78380110811524380991872173294, −0.65499976612598078142344493006,
1.08683781546487724886072393011, 2.76799530600685549694674607789, 3.16667747791961641456405065228, 4.03212878785980491716449351139, 5.22155060735590159892040235389, 5.79066634018440702026111782032, 6.52597300123134092999169353600, 7.54120191914906490894499715158, 8.115011261012247470513447212755, 8.823879534842734151104869138893
