| L(s) = 1 | + 1.41i·5-s + 2.44i·7-s + (1.73 − 2.82i)11-s − 4.89i·13-s − 7.34i·19-s + 2.82i·23-s + 2.99·25-s + 6.92·29-s + 4·31-s − 3.46·35-s + 8·37-s − 6.92·41-s + 2.44i·43-s + 2.82i·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s + 0.925i·7-s + (0.522 − 0.852i)11-s − 1.35i·13-s − 1.68i·19-s + 0.589i·23-s + 0.599·25-s + 1.28·29-s + 0.718·31-s − 0.585·35-s + 1.31·37-s − 1.08·41-s + 0.373i·43-s + 0.412i·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782793259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782793259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-1.73 + 2.82i)T \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 4.89iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.296981774932015322939652783521, −8.680463449611245591391335995649, −7.924870628634696348207911677879, −6.92450313982511848113281293585, −6.16828252102104571580763859557, −5.44814185830142154316644482154, −4.46013737402614262524647188469, −2.95326940179145000830286524812, −2.80867855613214972286391302053, −0.913581797576034709342425983506,
1.05757970338854818350780474448, 2.08099096636915018248600525359, 3.69012820238815085858315008773, 4.36737209730871194374599747372, 5.05286034888185456824213993074, 6.47832881978869266217410491339, 6.80309582067324016651017445986, 7.938764869339594152019472464919, 8.537794703256386964712700446764, 9.560533097538488635099731721504
