| L(s) = 1 | − 1.80i·3-s + (1.73 − 1.41i)5-s + 2.12·7-s − 0.267·9-s − 4.11i·11-s + 2.72·13-s + (−2.55 − 3.13i)15-s − 1.79i·17-s + (−3.49 + 2.60i)19-s − 3.85i·21-s + 3.68·23-s + (0.999 − 4.89i)25-s − 4.93i·27-s + 10.5i·29-s − 4.42·31-s + ⋯ |
| L(s) = 1 | − 1.04i·3-s + (0.774 − 0.632i)5-s + 0.804·7-s − 0.0893·9-s − 1.24i·11-s + 0.755·13-s + (−0.660 − 0.808i)15-s − 0.434i·17-s + (−0.801 + 0.598i)19-s − 0.840i·21-s + 0.769·23-s + (0.199 − 0.979i)25-s − 0.950i·27-s + 1.95i·29-s − 0.795·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287291096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287291096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
| 19 | \( 1 + (3.49 - 2.60i)T \) |
| good | 3 | \( 1 + 1.80iT - 3T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 + 4.11iT - 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 - 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.85iT - 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 4.42T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 3.13iT - 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 10.5iT - 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.929369724601429460985184431604, −8.472443880937525129162787938636, −7.72917644160899085570675065384, −6.73361474359276913154556366223, −6.00736688818271539443267669919, −5.28902368997003601225020791300, −4.24516832743135548898087677620, −2.88822513908441519794906450133, −1.64652741754551129490909790766, −1.01855833351475037915899528589,
1.65458826878787474261546191333, 2.63131863652603592530661260947, 4.00423790998016043125373105882, 4.54049791235048855680951102273, 5.48604704290825283369250433360, 6.38594110567800435541829296317, 7.26102245760994487200738233773, 8.161717724689157409635650580848, 9.256497911744253927126856977300, 9.632036010558451384421290538972
