| L(s) = 1 | − i·3-s − 4.37i·5-s − 2i·7-s − 9-s − 2.37i·11-s + 0.372·13-s − 4.37·15-s + (−3.37 + 2.37i)17-s − 2.37·19-s − 2·21-s − 4.37i·23-s − 14.1·25-s + i·27-s − 2i·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.95i·5-s − 0.755i·7-s − 0.333·9-s − 0.715i·11-s + 0.103·13-s − 1.12·15-s + (−0.817 + 0.575i)17-s − 0.544·19-s − 0.436·21-s − 0.911i·23-s − 2.82·25-s + 0.192i·27-s − 0.371i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9911391311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9911391311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 17 | \( 1 + (3.37 - 2.37i)T \) |
| good | 5 | \( 1 + 4.37iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 13 | \( 1 - 0.372T + 13T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 4.37iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 7.11iT - 41T^{2} \) |
| 43 | \( 1 + 1.62T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 + 14.7iT - 71T^{2} \) |
| 73 | \( 1 + 4.74iT - 73T^{2} \) |
| 79 | \( 1 + 10.7iT - 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.221081241126151559551710302314, −7.66944407670832547341841099604, −6.50522012896487713858348416616, −5.99984503627106048784719456552, −4.91887359123569354450521760252, −4.45446001302471120010327606758, −3.52877731046838866656755745908, −2.08120416860859113664900348462, −1.15045857952754442354123567460, −0.31511607889166267362731035731,
2.18919010034570984198668506113, 2.60055023310364210873200887107, 3.62025068111927228971358644391, 4.32153048206210982654651596321, 5.54030335542709490328521297544, 6.06981196290276167201513333662, 7.05540265225287955732584575558, 7.33995871081414736920499194904, 8.450767912925400319450118793605, 9.313774570399960276436961583407
