L(s) = 1 | − 2.82i·5-s − 4·7-s − 1.41i·11-s + 2.82i·13-s + 4·17-s + 7.07i·19-s + 4·23-s − 3.00·25-s − 8.48i·29-s + 8·31-s + 11.3i·35-s − 2.82i·37-s + 2·41-s + 4.24i·43-s + 9·49-s + ⋯ |
L(s) = 1 | − 1.26i·5-s − 1.51·7-s − 0.426i·11-s + 0.784i·13-s + 0.970·17-s + 1.62i·19-s + 0.834·23-s − 0.600·25-s − 1.57i·29-s + 1.43·31-s + 1.91i·35-s − 0.464i·37-s + 0.312·41-s + 0.646i·43-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.371052496\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371052496\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.07iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.189441084571430024637240307571, −7.55821070530495300811118391702, −6.41153081886503360947826690510, −6.07880725083413943609522241266, −5.23707400733695520539965478785, −4.31924812740243722322705975852, −3.63363505598303257181130820558, −2.77201676326465102970385078192, −1.46516309687750156498931043706, −0.49092480687420481332913385053,
0.919408787043590471133176061881, 2.76992467938964217704428403515, 2.86541428627448732923825248206, 3.68230855719850684798087509356, 4.86624119400846738332979582099, 5.68809928763986663219427487709, 6.58129745261416527557122659177, 6.93451179483899822699887599326, 7.47981042787439762883515221377, 8.568080518849351053973754922393