L(s) = 1 | + i·3-s + i·7-s − 9-s + 2·11-s + 4i·13-s − 2i·17-s + 2·19-s − 21-s + 4i·23-s − i·27-s + 2·29-s − 6·31-s + 2i·33-s + 6i·37-s − 4·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 1.10i·13-s − 0.485i·17-s + 0.458·19-s − 0.218·21-s + 0.834i·23-s − 0.192i·27-s + 0.371·29-s − 1.07·31-s + 0.348i·33-s + 0.986i·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.480134417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480134417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.144593807832547590411866122455, −8.996631486353940443550737405198, −7.79096866954398236662650254631, −7.01724407870329187713910140161, −6.14750204015814133902886614277, −5.33156950713950827684392588529, −4.46044355146702724611483695480, −3.67820661039193682650115059487, −2.65150396854639750385185438266, −1.41814933429407969278404174820,
0.54326065222918145208496200978, 1.72292186223764928279766773872, 2.93294280468274022841632116872, 3.81679617722819951054387158047, 4.86500582160276436637647678343, 5.84212625150835119824347244565, 6.47127456969836905409128006463, 7.42835712050021839219411176940, 7.913947825464193298343188648119, 8.815961489863402958343030494724