| L(s) = 1 | + (1 + i)3-s + (−3 − 4i)5-s + (−7 + 7i)7-s − 7i·9-s + 10·11-s + (9 + 9i)13-s + (1 − 7i)15-s + (1 − i)17-s + 8i·19-s − 14·21-s + (−23 − 23i)23-s + (−7 + 24i)25-s + (16 − 16i)27-s + 8i·29-s − 14·31-s + ⋯ |
| L(s) = 1 | + (0.333 + 0.333i)3-s + (−0.600 − 0.800i)5-s + (−1 + i)7-s − 0.777i·9-s + 0.909·11-s + (0.692 + 0.692i)13-s + (0.0666 − 0.466i)15-s + (0.0588 − 0.0588i)17-s + 0.421i·19-s − 0.666·21-s + (−1 − i)23-s + (−0.280 + 0.959i)25-s + (0.592 − 0.592i)27-s + 0.275i·29-s − 0.451·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.855497 + 0.0384920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.855497 + 0.0384920i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3 + 4i)T \) |
| good | 3 | \( 1 + (-1 - i)T + 9iT^{2} \) |
| 7 | \( 1 + (7 - 7i)T - 49iT^{2} \) |
| 11 | \( 1 - 10T + 121T^{2} \) |
| 13 | \( 1 + (-9 - 9i)T + 169iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (23 + 23i)T + 529iT^{2} \) |
| 29 | \( 1 - 8iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 + (-33 + 33i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + (15 + 15i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (39 - 39i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (7 + 7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 56iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (7 - 7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 98T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49 - 49i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 96iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63 + 63i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 112iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33 + 33i)T - 9.40e3iT^{2} \) |
| 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
−18.35124538472176511757358714269, −16.54259600717954489917133610160, −15.77141160655571916844795700460, −14.50339807607041166725982042675, −12.68128665727435992573424581356, −11.78936999005442889405940598420, −9.494964812819170688883265412784, −8.664380106294927574380010983587, −6.27462099292704690001749659643, −3.85769210437284222679930271984,
3.58149732306835412927525427068, 6.64188074165720645380338848513, 7.931175664981236379641946395835, 10.02152727589583107599767457175, 11.32836035526420098193819171841, 13.13258781276970670552049700998, 14.10552965086306807216091377726, 15.61439987312263760175240269667, 16.79087467384288582269870954105, 18.34748371195793641274209620802
