L(s) = 1 | − i·2-s − 4-s + i·8-s − 4·11-s − i·13-s + 16-s + 6i·17-s − 4·19-s + 4i·22-s + 8i·23-s − 26-s + 6·29-s − 8·31-s − i·32-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.353i·8-s − 1.20·11-s − 0.277i·13-s + 0.250·16-s + 1.45i·17-s − 0.917·19-s + 0.852i·22-s + 1.66i·23-s − 0.196·26-s + 1.11·29-s − 1.43·31-s − 0.176i·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.012050612\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.012050612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81334780575291103524175800491, −7.48152170200871643282777105370, −6.26791664491462523910704401979, −5.62036225757663055686889139876, −4.98373900124573529542895096691, −3.96175539939132962797576294529, −3.45101078527810027276363554517, −2.36343578402863494635925157362, −1.73578775414940570805044367105, −0.33460377106880606640827181061,
0.814609396860924717355402228032, 2.38714043190184018090325585699, 2.94880541408032256572315728980, 4.27295905786050973221422936989, 4.73733487755462747022174169327, 5.48428650712025179587996566552, 6.28324191211486244927667813695, 6.93914798720800732800545544368, 7.56849210619634473157221399917, 8.286258709119107122630602315765