L(s) = 1 | + 2.82i·5-s − 1.41i·7-s − 2·13-s − 1.41i·17-s + 2.82i·19-s + 6·23-s − 3.00·25-s − 5.65i·29-s − 9.89i·31-s + 4.00·35-s + 2·37-s + 4.24i·41-s + 8.48i·43-s + 10·47-s + 5·49-s + ⋯ |
L(s) = 1 | + 1.26i·5-s − 0.534i·7-s − 0.554·13-s − 0.342i·17-s + 0.648i·19-s + 1.25·23-s − 0.600·25-s − 1.05i·29-s − 1.77i·31-s + 0.676·35-s + 0.328·37-s + 0.662i·41-s + 1.29i·43-s + 1.45·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.804818769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804818769\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 9.89iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 12.7iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.243407613607398816545171528552, −7.47226591349202500757507217901, −7.16182202388232032733600562585, −6.28344220831121769811173796880, −5.68554834163475471066921613941, −4.55579746025051230656506941528, −3.90067216209622636299953080181, −2.89387165350341469181261838315, −2.36503840258358066388202104257, −0.892789921136727431852781688924,
0.64711876259644467384693553227, 1.67080124538445968311198073427, 2.71322142207489615024343636441, 3.67320386780717945841491230456, 4.79630525312801323286761518690, 5.07564062504294069323065310653, 5.82619666635213650403729242706, 6.91035365787776931389898515052, 7.40308398296919935030251549440, 8.541348756136689073782896055379