L(s) = 1 | + (−0.655 + 2.13i)5-s + (2.09 + 1.61i)7-s + 6.16i·11-s − 0.742·13-s + 3.80i·17-s + 7.08i·19-s − 5.47·23-s + (−4.13 − 2.80i)25-s − 3.48i·29-s − 7.72i·31-s + (−4.82 + 3.41i)35-s − 4.20i·37-s − 1.52·41-s − 12.0i·43-s − 0.165i·47-s + ⋯ |
L(s) = 1 | + (−0.293 + 0.956i)5-s + (0.791 + 0.611i)7-s + 1.85i·11-s − 0.205·13-s + 0.924i·17-s + 1.62i·19-s − 1.14·23-s + (−0.827 − 0.560i)25-s − 0.646i·29-s − 1.38i·31-s + (−0.816 + 0.577i)35-s − 0.691i·37-s − 0.238·41-s − 1.83i·43-s − 0.0240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.316939101\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316939101\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.655 - 2.13i)T \) |
| 7 | \( 1 + (-2.09 - 1.61i)T \) |
good | 11 | \( 1 - 6.16iT - 11T^{2} \) |
| 13 | \( 1 + 0.742T + 13T^{2} \) |
| 17 | \( 1 - 3.80iT - 17T^{2} \) |
| 19 | \( 1 - 7.08iT - 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 3.48iT - 29T^{2} \) |
| 31 | \( 1 + 7.72iT - 31T^{2} \) |
| 37 | \( 1 + 4.20iT - 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 0.165iT - 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.03iT - 61T^{2} \) |
| 67 | \( 1 - 0.383iT - 67T^{2} \) |
| 71 | \( 1 + 5.81iT - 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 - 0.779T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−9.352835865108406106993102804442, −8.214172799796539899107012135250, −7.77157074194178661940695744303, −7.09841098229062380433850594139, −6.10931972837971091065166572314, −5.48770398900907460974011296733, −4.24879929080154803699736189743, −3.84350092126321084531965808822, −2.23034756450956868263203581380, −1.95845673054635494807134043329,
0.44602221012018338931663343257, 1.32222459236086803256942344558, 2.82291151451778271866263288839, 3.74919329616879391099087152007, 4.80286873203492373844822487449, 5.14831781659349917422482147748, 6.22745664738865508836224450839, 7.15081172211001344394645764284, 7.970964804302422024632614106186, 8.569131076313538028376489265517