| L(s) = 1 | + 4i·3-s + (3 + 10.7i)5-s − 4i·7-s + 11·9-s + 43.0·11-s + 21.5i·13-s + (−43.0 + 12i)15-s + 43.0i·17-s + 129.·19-s + 16·21-s − 52i·23-s + (−106. + 64.6i)25-s + 152i·27-s − 158·29-s − 172.·31-s + ⋯ |
| L(s) = 1 | + 0.769i·3-s + (0.268 + 0.963i)5-s − 0.215i·7-s + 0.407·9-s + 1.18·11-s + 0.459i·13-s + (−0.741 + 0.206i)15-s + 0.614i·17-s + 1.56·19-s + 0.166·21-s − 0.471i·23-s + (−0.855 + 0.516i)25-s + 1.08i·27-s − 1.01·29-s − 0.998·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.102765095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.102765095\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3 - 10.7i)T \) |
| good | 3 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 + 4iT - 343T^{2} \) |
| 11 | \( 1 - 43.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 43.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 158T + 2.43e4T^{2} \) |
| 31 | \( 1 + 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 280. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 170T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 244iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 495. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 646.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82T + 2.26e5T^{2} \) |
| 67 | \( 1 - 692iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 344.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 940iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3iT - 9.12e5T^{2} \) |
| 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
−11.30790819982174645084270036262, −10.46476908877465645302359531877, −9.706746487669844775187820393853, −8.981579919983805669645477154371, −7.42493460130815172522521486026, −6.71465616967036987419443743749, −5.52634749362905251221890284059, −4.13948253143165837337601582924, −3.36492477554099515696462360290, −1.61640797696914413981665877773,
0.842347184336042013070015873529, 1.83517951677057620329067047007, 3.64395668175804502055556935066, 5.01346302371628956398508410082, 5.98266567921707765269299099833, 7.17571904774894764788738321438, 7.929878206461381410977859231740, 9.275064262436108460850434415568, 9.574377270996392113911424635951, 11.17990961661716073109714249680
